4.6 applied optimization - dr. travers page of...
TRANSCRIPT
![Page 1: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/1.jpg)
§ 4.6 Applied Optimization
![Page 2: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/2.jpg)
Practical Tips
1 Make sure you know what quantity is being optimized.
2 Sketch a diagram of the situation and label all variables in thesketch.
3 Obtain a formula for the quantity to be optimized and eliminateall variables but one (where applicable). Find the domain.
4 Find the critical points and evaluate at these points and theendpoints to find the global minimum or maximum, whichever isneeded.
![Page 3: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/3.jpg)
Practical Tips
1 Make sure you know what quantity is being optimized.2 Sketch a diagram of the situation and label all variables in the
sketch.
3 Obtain a formula for the quantity to be optimized and eliminateall variables but one (where applicable). Find the domain.
4 Find the critical points and evaluate at these points and theendpoints to find the global minimum or maximum, whichever isneeded.
![Page 4: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/4.jpg)
Practical Tips
1 Make sure you know what quantity is being optimized.2 Sketch a diagram of the situation and label all variables in the
sketch.3 Obtain a formula for the quantity to be optimized and eliminate
all variables but one (where applicable). Find the domain.
4 Find the critical points and evaluate at these points and theendpoints to find the global minimum or maximum, whichever isneeded.
![Page 5: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/5.jpg)
Practical Tips
1 Make sure you know what quantity is being optimized.2 Sketch a diagram of the situation and label all variables in the
sketch.3 Obtain a formula for the quantity to be optimized and eliminate
all variables but one (where applicable). Find the domain.4 Find the critical points and evaluate at these points and the
endpoints to find the global minimum or maximum, whichever isneeded.
![Page 6: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/6.jpg)
Fencing Example
Example
You want to fence a rectangular region of area 1000 ft2. You choosetwo different kinds of fencing to use: the parallel sides on the frontand back of the region will be fenced with $5 per foot fencing. Theleft and right sides will cost $3 per foot. Minimize the cost.
What quantity are we trying to optimize?
Is this a minimization or maximization problem?
What is our domain? 0 < x <∞
![Page 7: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/7.jpg)
Fencing Example
Example
You want to fence a rectangular region of area 1000 ft2. You choosetwo different kinds of fencing to use: the parallel sides on the frontand back of the region will be fenced with $5 per foot fencing. Theleft and right sides will cost $3 per foot. Minimize the cost.
What quantity are we trying to optimize?
Is this a minimization or maximization problem?
What is our domain? 0 < x <∞
![Page 8: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/8.jpg)
Fencing Example
Example
You want to fence a rectangular region of area 1000 ft2. You choosetwo different kinds of fencing to use: the parallel sides on the frontand back of the region will be fenced with $5 per foot fencing. Theleft and right sides will cost $3 per foot. Minimize the cost.
What quantity are we trying to optimize?
Is this a minimization or maximization problem?
What is our domain? 0 < x <∞
![Page 9: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/9.jpg)
Fencing Example
Example
You want to fence a rectangular region of area 1000 ft2. You choosetwo different kinds of fencing to use: the parallel sides on the frontand back of the region will be fenced with $5 per foot fencing. Theleft and right sides will cost $3 per foot. Minimize the cost.
What quantity are we trying to optimize?
Is this a minimization or maximization problem?
What is our domain?
0 < x <∞
![Page 10: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/10.jpg)
Fencing Example
Example
You want to fence a rectangular region of area 1000 ft2. You choosetwo different kinds of fencing to use: the parallel sides on the frontand back of the region will be fenced with $5 per foot fencing. Theleft and right sides will cost $3 per foot. Minimize the cost.
What quantity are we trying to optimize?
Is this a minimization or maximization problem?
What is our domain? 0 < x <∞
![Page 11: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/11.jpg)
Fencing Example
Now we need to visualize the situation.
5x
3yA = xy = 1000 ft2
C=10x+6y
![Page 12: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/12.jpg)
Fencing Example
Now we need to visualize the situation.
5x
3yA = xy = 1000 ft2
C=10x+6y
![Page 13: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/13.jpg)
Fencing Example
Now we need to visualize the situation.
5x
3yA = xy = 1000 ft2
C=10x+6y
![Page 14: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/14.jpg)
Fencing Example
Now we need to visualize the situation.
5x
3y
A = xy = 1000 ft2
C=10x+6y
![Page 15: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/15.jpg)
Fencing Example
Now we need to visualize the situation.
5x
3yA = xy = 1000 ft2
C=10x+6y
![Page 16: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/16.jpg)
Fencing Example
Now we need to visualize the situation.
5x
3yA = xy = 1000 ft2
C=10x+6y
![Page 17: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/17.jpg)
Fencing Example
We have two variables, which is why we need the two equations tosolve. What can we do?
We need to rewrite one equation to be able to write the other equationin terms of only one variable. Which way should we approach this?
A = xy = 1000
C = 10x + 6y
xy = 1000
y =1000
x
![Page 18: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/18.jpg)
Fencing Example
We have two variables, which is why we need the two equations tosolve. What can we do?
We need to rewrite one equation to be able to write the other equationin terms of only one variable. Which way should we approach this?
A = xy = 1000
C = 10x + 6y
xy = 1000
y =1000
x
![Page 19: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/19.jpg)
Fencing Example
We have two variables, which is why we need the two equations tosolve. What can we do?
We need to rewrite one equation to be able to write the other equationin terms of only one variable. Which way should we approach this?
A = xy = 1000
C = 10x + 6y
xy = 1000
y =1000
x
![Page 20: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/20.jpg)
Fencing Example
We have two variables, which is why we need the two equations tosolve. What can we do?
We need to rewrite one equation to be able to write the other equationin terms of only one variable. Which way should we approach this?
A = xy = 1000
C = 10x + 6y
xy = 1000
y =1000
x
![Page 21: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/21.jpg)
Fencing Example
C = 10x + 6y
C = 10x +6000
x
C′ = 10− 6000x2 = 0
10 =6000
x2
x2 = 600
x =√
600 ≈ 24.49 ft
![Page 22: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/22.jpg)
Fencing Example
C = 10x + 6y
C = 10x +6000
x
C′ = 10− 6000x2 = 0
10 =6000
x2
x2 = 600
x =√
600 ≈ 24.49 ft
![Page 23: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/23.jpg)
Fencing Example
C = 10x + 6y
C = 10x +6000
x
C′ =
10− 6000x2 = 0
10 =6000
x2
x2 = 600
x =√
600 ≈ 24.49 ft
![Page 24: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/24.jpg)
Fencing Example
C = 10x + 6y
C = 10x +6000
x
C′ = 10− 6000x2
= 0
10 =6000
x2
x2 = 600
x =√
600 ≈ 24.49 ft
![Page 25: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/25.jpg)
Fencing Example
C = 10x + 6y
C = 10x +6000
x
C′ = 10− 6000x2 = 0
10 =6000
x2
x2 = 600
x =√
600 ≈ 24.49 ft
![Page 26: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/26.jpg)
Fencing Example
C = 10x + 6y
C = 10x +6000
x
C′ = 10− 6000x2 = 0
10 =6000
x2
x2 = 600
x =√
600 ≈ 24.49 ft
![Page 27: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/27.jpg)
Fencing Example
C = 10x + 6y
C = 10x +6000
x
C′ = 10− 6000x2 = 0
10 =6000
x2
x2 = 600
x =√
600 ≈ 24.49 ft
![Page 28: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/28.jpg)
Fencing Example
C = 10x + 6y
C = 10x +6000
x
C′ = 10− 6000x2 = 0
10 =6000
x2
x2 = 600
x =√
600 ≈ 24.49
ft
![Page 29: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/29.jpg)
Fencing Example
C = 10x + 6y
C = 10x +6000
x
C′ = 10− 6000x2 = 0
10 =6000
x2
x2 = 600
x =√
600 ≈ 24.49 ft
![Page 30: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/30.jpg)
Fencing Example
Does this give us a minimum cost? How can we figure that out?
C′ = 10− 6000x2
C′′ =12000
x3
C′′∣∣x=24.49 > 0
So, we have a local minimum. Is this the only value we need toconsider?
![Page 31: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/31.jpg)
Fencing Example
Does this give us a minimum cost? How can we figure that out?
C′ = 10− 6000x2
C′′ =12000
x3
C′′∣∣x=24.49 > 0
So, we have a local minimum. Is this the only value we need toconsider?
![Page 32: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/32.jpg)
Fencing Example
Does this give us a minimum cost? How can we figure that out?
C′ = 10− 6000x2
C′′ =12000
x3
C′′∣∣x=24.49 > 0
So, we have a local minimum. Is this the only value we need toconsider?
![Page 33: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/33.jpg)
Fencing Example
Does this give us a minimum cost? How can we figure that out?
C′ = 10− 6000x2
C′′ =12000
x3
C′′∣∣x=24.49 > 0
So, we have a local minimum. Is this the only value we need toconsider?
![Page 34: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/34.jpg)
Fencing Example
Does this give us a minimum cost? How can we figure that out?
C′ = 10− 6000x2
C′′ =12000
x3
C′′∣∣x=24.49 > 0
So, we have a local minimum. Is this the only value we need toconsider?
![Page 35: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/35.jpg)
Fencing Example
Is this x value what we need?
We need the cost ...
C = 10(24.49) +600024.49
= $489.90
![Page 36: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/36.jpg)
Fencing Example
Is this x value what we need?
We need the cost ...
C = 10(24.49) +600024.49
= $489.90
![Page 37: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/37.jpg)
Fencing Example
Is this x value what we need?
We need the cost ...
C = 10(24.49) +600024.49
= $489.90
![Page 38: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/38.jpg)
Fencing Example
Is this x value what we need?
We need the cost ...
C = 10(24.49) +600024.49
= $489.90
![Page 39: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/39.jpg)
Circular Cone Example
ExampleFind the volume of the largest right circular cone that can be inscribedin a sphere of radius 19.
What quantity are we trying to optimize?
We want to maximize the volume of the cone. What is the volume ofa right circular cone?
The volume of a right circular cone is 13πr2h where r = 19 units.
What is the domain for the radius? 0 ≤ r ≤ 19
![Page 40: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/40.jpg)
Circular Cone Example
ExampleFind the volume of the largest right circular cone that can be inscribedin a sphere of radius 19.
What quantity are we trying to optimize?
We want to maximize the volume of the cone. What is the volume ofa right circular cone?
The volume of a right circular cone is 13πr2h where r = 19 units.
What is the domain for the radius? 0 ≤ r ≤ 19
![Page 41: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/41.jpg)
Circular Cone Example
ExampleFind the volume of the largest right circular cone that can be inscribedin a sphere of radius 19.
What quantity are we trying to optimize?
We want to maximize the volume of the cone. What is the volume ofa right circular cone?
The volume of a right circular cone is 13πr2h where r = 19 units.
What is the domain for the radius? 0 ≤ r ≤ 19
![Page 42: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/42.jpg)
Circular Cone Example
ExampleFind the volume of the largest right circular cone that can be inscribedin a sphere of radius 19.
What quantity are we trying to optimize?
We want to maximize the volume of the cone. What is the volume ofa right circular cone?
The volume of a right circular cone is 13πr2h where r = 19 units.
What is the domain for the radius? 0 ≤ r ≤ 19
![Page 43: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/43.jpg)
Circular Cone Example
ExampleFind the volume of the largest right circular cone that can be inscribedin a sphere of radius 19.
What quantity are we trying to optimize?
We want to maximize the volume of the cone. What is the volume ofa right circular cone?
The volume of a right circular cone is 13πr2h where r = 19 units.
What is the domain for the radius?
0 ≤ r ≤ 19
![Page 44: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/44.jpg)
Circular Cone Example
ExampleFind the volume of the largest right circular cone that can be inscribedin a sphere of radius 19.
What quantity are we trying to optimize?
We want to maximize the volume of the cone. What is the volume ofa right circular cone?
The volume of a right circular cone is 13πr2h where r = 19 units.
What is the domain for the radius? 0 ≤ r ≤ 19
![Page 45: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/45.jpg)
Circular Cone Example
We need to now visualize the situation.
![Page 46: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/46.jpg)
Circular Cone Example
We need to now visualize the situation.
![Page 47: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/47.jpg)
Circular Cone Example
Let s be the radius of the cone. We have that r is the hypotenuse of theright triangle. What equation can we write to relate the value of x tothe rest of the problem?
x = h− 19 .
Now? By the Pythagorean Theorem,
s2 = 192 − x2 = 361− x2
So, the radius is a function of x. We want the same for the height ofthe cone because we need one variable. What is this equation?h = 19 + x .
![Page 48: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/48.jpg)
Circular Cone Example
Let s be the radius of the cone. We have that r is the hypotenuse of theright triangle. What equation can we write to relate the value of x tothe rest of the problem? x = h− 19 .
Now? By the Pythagorean Theorem,
s2 = 192 − x2 = 361− x2
So, the radius is a function of x. We want the same for the height ofthe cone because we need one variable. What is this equation?h = 19 + x .
![Page 49: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/49.jpg)
Circular Cone Example
Let s be the radius of the cone. We have that r is the hypotenuse of theright triangle. What equation can we write to relate the value of x tothe rest of the problem? x = h− 19 .
Now?
By the Pythagorean Theorem,
s2 = 192 − x2 = 361− x2
So, the radius is a function of x. We want the same for the height ofthe cone because we need one variable. What is this equation?h = 19 + x .
![Page 50: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/50.jpg)
Circular Cone Example
Let s be the radius of the cone. We have that r is the hypotenuse of theright triangle. What equation can we write to relate the value of x tothe rest of the problem? x = h− 19 .
Now? By the Pythagorean Theorem,
s2 = 192 − x2 = 361− x2
So, the radius is a function of x. We want the same for the height ofthe cone because we need one variable. What is this equation?h = 19 + x .
![Page 51: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/51.jpg)
Circular Cone Example
Let s be the radius of the cone. We have that r is the hypotenuse of theright triangle. What equation can we write to relate the value of x tothe rest of the problem? x = h− 19 .
Now? By the Pythagorean Theorem,
s2 = 192 − x2 = 361− x2
So, the radius is a function of x. We want the same for the height ofthe cone because we need one variable. What is this equation?
h = 19 + x .
![Page 52: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/52.jpg)
Circular Cone Example
Let s be the radius of the cone. We have that r is the hypotenuse of theright triangle. What equation can we write to relate the value of x tothe rest of the problem? x = h− 19 .
Now? By the Pythagorean Theorem,
s2 = 192 − x2 = 361− x2
So, the radius is a function of x. We want the same for the height ofthe cone because we need one variable. What is this equation?h = 19 + x .
![Page 53: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/53.jpg)
Circular Cone Example
Now,
V =13πr2h
=π
3(361− x2)(19 + x)
=π
3(193 + 192x− 19x2 − x3)
Now what?
V ′ =π
3(192 − 38x− 3x2) = 0
π
3(19 + x)(19− 3x) = 0
x = −19,193
The negative root is impossible, so we have only one option.
![Page 54: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/54.jpg)
Circular Cone Example
Now,
V =13πr2h
=π
3(361− x2)(19 + x)
=π
3(193 + 192x− 19x2 − x3)
Now what?
V ′ =π
3(192 − 38x− 3x2) = 0
π
3(19 + x)(19− 3x) = 0
x = −19,193
The negative root is impossible, so we have only one option.
![Page 55: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/55.jpg)
Circular Cone Example
Now,
V =13πr2h
=π
3(361− x2)(19 + x)
=π
3(193 + 192x− 19x2 − x3)
Now what?
V ′ =π
3(192 − 38x− 3x2) = 0
π
3(19 + x)(19− 3x) = 0
x = −19,193
The negative root is impossible, so we have only one option.
![Page 56: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/56.jpg)
Circular Cone Example
Now,
V =13πr2h
=π
3(361− x2)(19 + x)
=π
3(193 + 192x− 19x2 − x3)
Now what?
V ′ =π
3(192 − 38x− 3x2) = 0
π
3(19 + x)(19− 3x) = 0
x = −19,193
The negative root is impossible, so we have only one option.
![Page 57: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/57.jpg)
Circular Cone Example
Now,
V =13πr2h
=π
3(361− x2)(19 + x)
=π
3(193 + 192x− 19x2 − x3)
Now what?
V ′ =π
3(192 − 38x− 3x2)
= 0π
3(19 + x)(19− 3x) = 0
x = −19,193
The negative root is impossible, so we have only one option.
![Page 58: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/58.jpg)
Circular Cone Example
Now,
V =13πr2h
=π
3(361− x2)(19 + x)
=π
3(193 + 192x− 19x2 − x3)
Now what?
V ′ =π
3(192 − 38x− 3x2) = 0
π
3(19 + x)(19− 3x) = 0
x = −19,193
The negative root is impossible, so we have only one option.
![Page 59: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/59.jpg)
Circular Cone Example
Now,
V =13πr2h
=π
3(361− x2)(19 + x)
=π
3(193 + 192x− 19x2 − x3)
Now what?
V ′ =π
3(192 − 38x− 3x2) = 0
π
3(19 + x)(19− 3x) = 0
x = −19,193
The negative root is impossible, so we have only one option.
![Page 60: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/60.jpg)
Circular Cone Example
Now,
V =13πr2h
=π
3(361− x2)(19 + x)
=π
3(193 + 192x− 19x2 − x3)
Now what?
V ′ =π
3(192 − 38x− 3x2) = 0
π
3(19 + x)(19− 3x) = 0
x = −19,193
The negative root is impossible, so we have only one option.
![Page 61: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/61.jpg)
Circular Cone Example
Now,
V =13πr2h
=π
3(361− x2)(19 + x)
=π
3(193 + 192x− 19x2 − x3)
Now what?
V ′ =π
3(192 − 38x− 3x2) = 0
π
3(19 + x)(19− 3x) = 0
x = −19,193
The negative root is impossible, so we have only one option.
![Page 62: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/62.jpg)
Circular Cone Example
What do we do with this?
V ′′ = − 38− 6x
V ′′∣∣x= 19
3< 0
Conclusion? r = 193 gives a local maximum.
![Page 63: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/63.jpg)
Circular Cone Example
What do we do with this?
V ′′ =
− 38− 6x
V ′′∣∣x= 19
3< 0
Conclusion? r = 193 gives a local maximum.
![Page 64: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/64.jpg)
Circular Cone Example
What do we do with this?
V ′′ = − 38− 6x
V ′′∣∣x= 19
3< 0
Conclusion? r = 193 gives a local maximum.
![Page 65: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/65.jpg)
Circular Cone Example
What do we do with this?
V ′′ = − 38− 6x
V ′′∣∣x= 19
3
< 0
Conclusion? r = 193 gives a local maximum.
![Page 66: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/66.jpg)
Circular Cone Example
What do we do with this?
V ′′ = − 38− 6x
V ′′∣∣x= 19
3< 0
Conclusion? r = 193 gives a local maximum.
![Page 67: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/67.jpg)
Circular Cone Example
What do we do with this?
V ′′ = − 38− 6x
V ′′∣∣x= 19
3< 0
Conclusion?
r = 193 gives a local maximum.
![Page 68: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/68.jpg)
Circular Cone Example
What do we do with this?
V ′′ = − 38− 6x
V ′′∣∣x= 19
3< 0
Conclusion? r = 193 gives a local maximum.
![Page 69: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/69.jpg)
Circular Cone Example
So is r = 193 units our answer?
V =π
3
(361− 361
9
)(19 +
193
)=
21948881
π ≈ 8512.86 units3.
![Page 70: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/70.jpg)
Circular Cone Example
So is r = 193 units our answer?
V =π
3
(361− 361
9
)(19 +
193
)=
21948881
π ≈ 8512.86 units3.
![Page 71: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/71.jpg)
Circular Cone Example
So is r = 193 units our answer?
V =π
3
(361− 361
9
)(19 +
193
)=
21948881
π ≈ 8512.86 units3.
![Page 72: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/72.jpg)
Circular Cylinder Example
ExampleFind the radius and height of the right circular cylinder of largestvolume that can be inscribed in a right circular cone with radius 6′′
and height 10′′.
What is the quantity we are optimizing?
We want to find the dimensions that maximize the volume of a rightcircular cylinder.
What geometric formulas do we need here?
Right circular cone: Vco = 13πr2h
Right circular cylinder: Vcy = πr2h
![Page 73: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/73.jpg)
Circular Cylinder Example
ExampleFind the radius and height of the right circular cylinder of largestvolume that can be inscribed in a right circular cone with radius 6′′
and height 10′′.
What is the quantity we are optimizing?
We want to find the dimensions that maximize the volume of a rightcircular cylinder.
What geometric formulas do we need here?
Right circular cone: Vco = 13πr2h
Right circular cylinder: Vcy = πr2h
![Page 74: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/74.jpg)
Circular Cylinder Example
ExampleFind the radius and height of the right circular cylinder of largestvolume that can be inscribed in a right circular cone with radius 6′′
and height 10′′.
What is the quantity we are optimizing?
We want to find the dimensions that maximize the volume of a rightcircular cylinder.
What geometric formulas do we need here?
Right circular cone: Vco = 13πr2h
Right circular cylinder: Vcy = πr2h
![Page 75: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/75.jpg)
Circular Cylinder Example
ExampleFind the radius and height of the right circular cylinder of largestvolume that can be inscribed in a right circular cone with radius 6′′
and height 10′′.
What is the quantity we are optimizing?
We want to find the dimensions that maximize the volume of a rightcircular cylinder.
What geometric formulas do we need here?
Right circular cone: Vco = 13πr2h
Right circular cylinder: Vcy = πr2h
![Page 76: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/76.jpg)
Circular Cylinder Example
ExampleFind the radius and height of the right circular cylinder of largestvolume that can be inscribed in a right circular cone with radius 6′′
and height 10′′.
What is the quantity we are optimizing?
We want to find the dimensions that maximize the volume of a rightcircular cylinder.
What geometric formulas do we need here?
Right circular cone:
Vco = 13πr2h
Right circular cylinder: Vcy = πr2h
![Page 77: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/77.jpg)
Circular Cylinder Example
ExampleFind the radius and height of the right circular cylinder of largestvolume that can be inscribed in a right circular cone with radius 6′′
and height 10′′.
What is the quantity we are optimizing?
We want to find the dimensions that maximize the volume of a rightcircular cylinder.
What geometric formulas do we need here?
Right circular cone: Vco = 13πr2h
Right circular cylinder: Vcy = πr2h
![Page 78: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/78.jpg)
Circular Cylinder Example
ExampleFind the radius and height of the right circular cylinder of largestvolume that can be inscribed in a right circular cone with radius 6′′
and height 10′′.
What is the quantity we are optimizing?
We want to find the dimensions that maximize the volume of a rightcircular cylinder.
What geometric formulas do we need here?
Right circular cone: Vco = 13πr2h
Right circular cylinder:
Vcy = πr2h
![Page 79: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/79.jpg)
Circular Cylinder Example
ExampleFind the radius and height of the right circular cylinder of largestvolume that can be inscribed in a right circular cone with radius 6′′
and height 10′′.
What is the quantity we are optimizing?
We want to find the dimensions that maximize the volume of a rightcircular cylinder.
What geometric formulas do we need here?
Right circular cone: Vco = 13πr2h
Right circular cylinder: Vcy = πr2h
![Page 80: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/80.jpg)
Circular Cylinder Example
What visual do we need here?
![Page 81: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/81.jpg)
Circular Cylinder Example
What visual do we need here?
![Page 82: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/82.jpg)
Circular Cylinder Example
Since there are two variables here (r and h) we need to find anotherrelationship between them. To do this we will use a cross- section.
10
6
r
h
10− h 106 = 10−h
r
h = − 53 r + 10
![Page 83: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/83.jpg)
Circular Cylinder Example
Since there are two variables here (r and h) we need to find anotherrelationship between them. To do this we will use a cross- section.
10
6
r
h
10− h 106 = 10−h
r
h = − 53 r + 10
![Page 84: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/84.jpg)
Circular Cylinder Example
Since there are two variables here (r and h) we need to find anotherrelationship between them. To do this we will use a cross- section.
10
6
r
h
10− h 106 = 10−h
r
h = − 53 r + 10
![Page 85: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/85.jpg)
Circular Cylinder Example
Since there are two variables here (r and h) we need to find anotherrelationship between them. To do this we will use a cross- section.
10
6
r
h
10− h 106 = 10−h
r
h = − 53 r + 10
![Page 86: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/86.jpg)
Circular Cylinder Example
Since there are two variables here (r and h) we need to find anotherrelationship between them. To do this we will use a cross- section.
10
6
r
h
10− h
106 = 10−h
r
h = − 53 r + 10
![Page 87: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/87.jpg)
Circular Cylinder Example
Since there are two variables here (r and h) we need to find anotherrelationship between them. To do this we will use a cross- section.
10
6
r
h
10− h 106 = 10−h
r
h = − 53 r + 10
![Page 88: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/88.jpg)
Circular Cylinder Example
Since there are two variables here (r and h) we need to find anotherrelationship between them. To do this we will use a cross- section.
10
6
r
h
10− h 106 = 10−h
r
h = − 53 r + 10
![Page 89: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/89.jpg)
Circular Cylinder Example
What is our domain for the radius here?
0 < r < 6
V = πr2(
10− 53
r)
= 10πr2 − 5π3
r3
dVdr
= 20πr − 5πr2 = 0
5πr(4− r) = 0
r = 0, 4
![Page 90: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/90.jpg)
Circular Cylinder Example
What is our domain for the radius here? 0 < r < 6
V = πr2(
10− 53
r)
= 10πr2 − 5π3
r3
dVdr
= 20πr − 5πr2 = 0
5πr(4− r) = 0
r = 0, 4
![Page 91: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/91.jpg)
Circular Cylinder Example
What is our domain for the radius here? 0 < r < 6
V = πr2(
10− 53
r)
=
10πr2 − 5π3
r3
dVdr
= 20πr − 5πr2 = 0
5πr(4− r) = 0
r = 0, 4
![Page 92: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/92.jpg)
Circular Cylinder Example
What is our domain for the radius here? 0 < r < 6
V = πr2(
10− 53
r)
= 10πr2 − 5π3
r3
dVdr
= 20πr − 5πr2 = 0
5πr(4− r) = 0
r = 0, 4
![Page 93: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/93.jpg)
Circular Cylinder Example
What is our domain for the radius here? 0 < r < 6
V = πr2(
10− 53
r)
= 10πr2 − 5π3
r3
dVdr
=
20πr − 5πr2 = 0
5πr(4− r) = 0
r = 0, 4
![Page 94: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/94.jpg)
Circular Cylinder Example
What is our domain for the radius here? 0 < r < 6
V = πr2(
10− 53
r)
= 10πr2 − 5π3
r3
dVdr
= 20πr − 5πr2
= 0
5πr(4− r) = 0
r = 0, 4
![Page 95: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/95.jpg)
Circular Cylinder Example
What is our domain for the radius here? 0 < r < 6
V = πr2(
10− 53
r)
= 10πr2 − 5π3
r3
dVdr
= 20πr − 5πr2 = 0
5πr(4− r) = 0
r = 0, 4
![Page 96: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/96.jpg)
Circular Cylinder Example
What is our domain for the radius here? 0 < r < 6
V = πr2(
10− 53
r)
= 10πr2 − 5π3
r3
dVdr
= 20πr − 5πr2 = 0
5πr(4− r) = 0
r = 0, 4
![Page 97: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/97.jpg)
Circular Cylinder Example
What is our domain for the radius here? 0 < r < 6
V = πr2(
10− 53
r)
= 10πr2 − 5π3
r3
dVdr
= 20πr − 5πr2 = 0
5πr(4− r) = 0
r = 0, 4
![Page 98: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/98.jpg)
Circular Cylinder Example
Now what do we need to do?
d2Vdr2 = 20π − 10πr
V ′′∣∣r=0 > 0
V ′′∣∣r=4 < 0
V ′′∣∣r=6 < 0
Conclusion? We have two local maxima. Which is the right one?
![Page 99: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/99.jpg)
Circular Cylinder Example
Now what do we need to do?
d2Vdr2 =
20π − 10πr
V ′′∣∣r=0 > 0
V ′′∣∣r=4 < 0
V ′′∣∣r=6 < 0
Conclusion? We have two local maxima. Which is the right one?
![Page 100: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/100.jpg)
Circular Cylinder Example
Now what do we need to do?
d2Vdr2 = 20π − 10πr
V ′′∣∣r=0 > 0
V ′′∣∣r=4 < 0
V ′′∣∣r=6 < 0
Conclusion? We have two local maxima. Which is the right one?
![Page 101: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/101.jpg)
Circular Cylinder Example
Now what do we need to do?
d2Vdr2 = 20π − 10πr
V ′′∣∣r=0
> 0
V ′′∣∣r=4 < 0
V ′′∣∣r=6 < 0
Conclusion? We have two local maxima. Which is the right one?
![Page 102: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/102.jpg)
Circular Cylinder Example
Now what do we need to do?
d2Vdr2 = 20π − 10πr
V ′′∣∣r=0 > 0
V ′′∣∣r=4
< 0
V ′′∣∣r=6 < 0
Conclusion? We have two local maxima. Which is the right one?
![Page 103: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/103.jpg)
Circular Cylinder Example
Now what do we need to do?
d2Vdr2 = 20π − 10πr
V ′′∣∣r=0 > 0
V ′′∣∣r=4 < 0
V ′′∣∣r=6
< 0
Conclusion? We have two local maxima. Which is the right one?
![Page 104: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/104.jpg)
Circular Cylinder Example
Now what do we need to do?
d2Vdr2 = 20π − 10πr
V ′′∣∣r=0 > 0
V ′′∣∣r=4 < 0
V ′′∣∣r=6 < 0
Conclusion? We have two local maxima. Which is the right one?
![Page 105: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/105.jpg)
Circular Cylinder Example
Now what do we need to do?
d2Vdr2 = 20π − 10πr
V ′′∣∣r=0 > 0
V ′′∣∣r=4 < 0
V ′′∣∣r=6 < 0
Conclusion?
We have two local maxima. Which is the right one?
![Page 106: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/106.jpg)
Circular Cylinder Example
Now what do we need to do?
d2Vdr2 = 20π − 10πr
V ′′∣∣r=0 > 0
V ′′∣∣r=4 < 0
V ′′∣∣r=6 < 0
Conclusion? We have two local maxima. Which is the right one?
![Page 107: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/107.jpg)
Circular Cylinder Example
V|r=6 =
360π − 360π = 0
V|r=4 = 160π − 320π3
=160π
3
So, we get the maximum volume of 160π3 when r = 4 and h = 10
3 .
![Page 108: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/108.jpg)
Circular Cylinder Example
V|r=6 = 360π − 360π = 0
V|r=4 = 160π − 320π3
=160π
3
So, we get the maximum volume of 160π3 when r = 4 and h = 10
3 .
![Page 109: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/109.jpg)
Circular Cylinder Example
V|r=6 = 360π − 360π = 0
V|r=4 =
160π − 320π3
=160π
3
So, we get the maximum volume of 160π3 when r = 4 and h = 10
3 .
![Page 110: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/110.jpg)
Circular Cylinder Example
V|r=6 = 360π − 360π = 0
V|r=4 = 160π − 320π3
=160π
3
So, we get the maximum volume of 160π3 when r = 4 and h = 10
3 .
![Page 111: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/111.jpg)
Circular Cylinder Example
V|r=6 = 360π − 360π = 0
V|r=4 = 160π − 320π3
=160π
3
So, we get the maximum volume of 160π3 when r = 4 and h = 10
3 .
![Page 112: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/112.jpg)
Shipping Example
ExampleBoise, Idaho is about 300 miles inland from the nearest point on thePacific Coast. San Diego is about 1000 miles south of that point downthe coast. Assuming the coast is a straight line going north to south, Cis the point on the coast directly west of Boise. It costs $.02 per mileto transport a ton of potatoes and $.01 to transport them by ship. TheIdaho Potato Company wants to find a point P, on the Pacific Coast,to which it should truck the potatoes before loading them aboard acargo ship in order to minimize cost.
Let’s start with a visual.
![Page 113: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/113.jpg)
Shipping Example
ExampleBoise, Idaho is about 300 miles inland from the nearest point on thePacific Coast. San Diego is about 1000 miles south of that point downthe coast. Assuming the coast is a straight line going north to south, Cis the point on the coast directly west of Boise. It costs $.02 per mileto transport a ton of potatoes and $.01 to transport them by ship. TheIdaho Potato Company wants to find a point P, on the Pacific Coast,to which it should truck the potatoes before loading them aboard acargo ship in order to minimize cost.
Let’s start with a visual.
![Page 114: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/114.jpg)
Shipping Example
300 mi
1000 mi
BC
SD
Px
1000− x√
x2 + 90000
![Page 115: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/115.jpg)
Shipping Example
300 mi
1000 mi
BC
SD
P
x
1000− x√
x2 + 90000
![Page 116: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/116.jpg)
Shipping Example
300 mi
1000 mi
BC
SD
P
x
1000− x√
x2 + 90000
![Page 117: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/117.jpg)
Shipping Example
300 mi
1000 mi
BC
SD
Px
1000− x√
x2 + 90000
![Page 118: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/118.jpg)
Shipping Example
300 mi
1000 mi
BC
SD
Px
1000− x
√x2 + 90000
![Page 119: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/119.jpg)
Shipping Example
300 mi
1000 mi
BC
SD
Px
1000− x
√x2 + 90000
![Page 120: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/120.jpg)
Shipping Example
300 mi
1000 mi
BC
SD
Px
1000− x√
x2 + 90000
![Page 121: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/121.jpg)
Shipping Example
What is the function we want to optimize?
D =√
x2 + 90000 + (1000− x)
C = 2√
x2 + 90000 + 1(1000− x)
Notice we didn’t use $.02 and $.01. This is because those decimalswould be harder to work with and the fact that the only thing thatreally matters is that it costs twice as much over land as it does byboat.
![Page 122: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/122.jpg)
Shipping Example
What is the function we want to optimize?
D =√
x2 + 90000 + (1000− x)
C = 2√
x2 + 90000 + 1(1000− x)
Notice we didn’t use $.02 and $.01. This is because those decimalswould be harder to work with and the fact that the only thing thatreally matters is that it costs twice as much over land as it does byboat.
![Page 123: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/123.jpg)
Shipping Example
What is the function we want to optimize?
D =√
x2 + 90000 + (1000− x)
C = 2√
x2 + 90000 + 1(1000− x)
Notice we didn’t use $.02 and $.01. This is because those decimalswould be harder to work with and the fact that the only thing thatreally matters is that it costs twice as much over land as it does byboat.
![Page 124: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/124.jpg)
Shipping Example
What is the function we want to optimize?
D =√
x2 + 90000 + (1000− x)
C = 2√
x2 + 90000 + 1(1000− x)
Notice we didn’t use $.02 and $.01. This is because those decimalswould be harder to work with and the fact that the only thing thatreally matters is that it costs twice as much over land as it does byboat.
![Page 125: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/125.jpg)
Shipping Example
C′ =
2x√x2 + 90000
− 1 = 0
2x√x2 + 90000
= 1
4x2
x2 + 90000= 1
4x2 = x2 + 90000
3x2 = 90000
x = ±173.21
The negative value is impossible, so we need only consider outdomain’s endpoints and the critical point.
![Page 126: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/126.jpg)
Shipping Example
C′ =2x√
x2 + 90000− 1
= 0
2x√x2 + 90000
= 1
4x2
x2 + 90000= 1
4x2 = x2 + 90000
3x2 = 90000
x = ±173.21
The negative value is impossible, so we need only consider outdomain’s endpoints and the critical point.
![Page 127: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/127.jpg)
Shipping Example
C′ =2x√
x2 + 90000− 1 = 0
2x√x2 + 90000
= 1
4x2
x2 + 90000= 1
4x2 = x2 + 90000
3x2 = 90000
x = ±173.21
The negative value is impossible, so we need only consider outdomain’s endpoints and the critical point.
![Page 128: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/128.jpg)
Shipping Example
C′ =2x√
x2 + 90000− 1 = 0
2x√x2 + 90000
= 1
4x2
x2 + 90000= 1
4x2 = x2 + 90000
3x2 = 90000
x = ±173.21
The negative value is impossible, so we need only consider outdomain’s endpoints and the critical point.
![Page 129: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/129.jpg)
Shipping Example
C′ =2x√
x2 + 90000− 1 = 0
2x√x2 + 90000
= 1
4x2
x2 + 90000= 1
4x2 = x2 + 90000
3x2 = 90000
x = ±173.21
The negative value is impossible, so we need only consider outdomain’s endpoints and the critical point.
![Page 130: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/130.jpg)
Shipping Example
C′ =2x√
x2 + 90000− 1 = 0
2x√x2 + 90000
= 1
4x2
x2 + 90000= 1
4x2 = x2 + 90000
3x2 = 90000
x = ±173.21
The negative value is impossible, so we need only consider outdomain’s endpoints and the critical point.
![Page 131: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/131.jpg)
Shipping Example
C′ =2x√
x2 + 90000− 1 = 0
2x√x2 + 90000
= 1
4x2
x2 + 90000= 1
4x2 = x2 + 90000
3x2 = 90000
x = ±173.21
The negative value is impossible, so we need only consider outdomain’s endpoints and the critical point.
![Page 132: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/132.jpg)
Shipping Example
C′ =2x√
x2 + 90000− 1 = 0
2x√x2 + 90000
= 1
4x2
x2 + 90000= 1
4x2 = x2 + 90000
3x2 = 90000
x = ±173.21
The negative value is impossible, so we need only consider outdomain’s endpoints and the critical point.
![Page 133: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/133.jpg)
Shipping Example
C′ =2x√
x2 + 90000− 1 = 0
2x√x2 + 90000
= 1
4x2
x2 + 90000= 1
4x2 = x2 + 90000
3x2 = 90000
x = ±173.21
The negative value is impossible, so we need only consider outdomain’s endpoints and the critical point.
![Page 134: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/134.jpg)
Shipping Example
What is the domain?
D(C) : 0 ≤ x ≤ 1000
What happens if we use x = 0? We are not cutting any travel out sincethat would entail the whole distance over land and the whole distanceover the ocean.
What happens if we use x = 1000? We are covering the entiredistance over land, which is the more expensive way to go and couldnot possibly minimize cost. So the endpoints won’t work.
Are we done?
We need to make sure the critical point is a minimum.
![Page 135: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/135.jpg)
Shipping Example
What is the domain? D(C) : 0 ≤ x ≤ 1000
What happens if we use x = 0? We are not cutting any travel out sincethat would entail the whole distance over land and the whole distanceover the ocean.
What happens if we use x = 1000? We are covering the entiredistance over land, which is the more expensive way to go and couldnot possibly minimize cost. So the endpoints won’t work.
Are we done?
We need to make sure the critical point is a minimum.
![Page 136: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/136.jpg)
Shipping Example
What is the domain? D(C) : 0 ≤ x ≤ 1000
What happens if we use x = 0?
We are not cutting any travel out sincethat would entail the whole distance over land and the whole distanceover the ocean.
What happens if we use x = 1000? We are covering the entiredistance over land, which is the more expensive way to go and couldnot possibly minimize cost. So the endpoints won’t work.
Are we done?
We need to make sure the critical point is a minimum.
![Page 137: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/137.jpg)
Shipping Example
What is the domain? D(C) : 0 ≤ x ≤ 1000
What happens if we use x = 0? We are not cutting any travel out sincethat would entail the whole distance over land and the whole distanceover the ocean.
What happens if we use x = 1000? We are covering the entiredistance over land, which is the more expensive way to go and couldnot possibly minimize cost. So the endpoints won’t work.
Are we done?
We need to make sure the critical point is a minimum.
![Page 138: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/138.jpg)
Shipping Example
What is the domain? D(C) : 0 ≤ x ≤ 1000
What happens if we use x = 0? We are not cutting any travel out sincethat would entail the whole distance over land and the whole distanceover the ocean.
What happens if we use x = 1000?
We are covering the entiredistance over land, which is the more expensive way to go and couldnot possibly minimize cost. So the endpoints won’t work.
Are we done?
We need to make sure the critical point is a minimum.
![Page 139: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/139.jpg)
Shipping Example
What is the domain? D(C) : 0 ≤ x ≤ 1000
What happens if we use x = 0? We are not cutting any travel out sincethat would entail the whole distance over land and the whole distanceover the ocean.
What happens if we use x = 1000? We are covering the entiredistance over land, which is the more expensive way to go and couldnot possibly minimize cost. So the endpoints won’t work.
Are we done?
We need to make sure the critical point is a minimum.
![Page 140: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/140.jpg)
Shipping Example
What is the domain? D(C) : 0 ≤ x ≤ 1000
What happens if we use x = 0? We are not cutting any travel out sincethat would entail the whole distance over land and the whole distanceover the ocean.
What happens if we use x = 1000? We are covering the entiredistance over land, which is the more expensive way to go and couldnot possibly minimize cost. So the endpoints won’t work.
Are we done?
We need to make sure the critical point is a minimum.
![Page 141: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/141.jpg)
Shipping Example
What is the domain? D(C) : 0 ≤ x ≤ 1000
What happens if we use x = 0? We are not cutting any travel out sincethat would entail the whole distance over land and the whole distanceover the ocean.
What happens if we use x = 1000? We are covering the entiredistance over land, which is the more expensive way to go and couldnot possibly minimize cost. So the endpoints won’t work.
Are we done?
We need to make sure the critical point is a minimum.
![Page 142: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/142.jpg)
Shipping Example
C′′ =
2√
x2 + 90000− 4x2√
x2+90000
x2 + 90000
=−2x2 + 180000
(x2 + 90000)32
C′′∣∣x=173.21 > 0
Conclusion? This point does give a local minimum. Therefore, thepoint P where the potatoes should be loaded onto a boat is at 173.21miles due south of point C.
![Page 143: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/143.jpg)
Shipping Example
C′′ =2√
x2 + 90000− 4x2√
x2+90000
x2 + 90000
=−2x2 + 180000
(x2 + 90000)32
C′′∣∣x=173.21 > 0
Conclusion? This point does give a local minimum. Therefore, thepoint P where the potatoes should be loaded onto a boat is at 173.21miles due south of point C.
![Page 144: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/144.jpg)
Shipping Example
C′′ =2√
x2 + 90000− 4x2√
x2+90000
x2 + 90000
=−2x2 + 180000
(x2 + 90000)32
C′′∣∣x=173.21 > 0
Conclusion? This point does give a local minimum. Therefore, thepoint P where the potatoes should be loaded onto a boat is at 173.21miles due south of point C.
![Page 145: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/145.jpg)
Shipping Example
C′′ =2√
x2 + 90000− 4x2√
x2+90000
x2 + 90000
=−2x2 + 180000
(x2 + 90000)32
C′′∣∣x=173.21
> 0
Conclusion? This point does give a local minimum. Therefore, thepoint P where the potatoes should be loaded onto a boat is at 173.21miles due south of point C.
![Page 146: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/146.jpg)
Shipping Example
C′′ =2√
x2 + 90000− 4x2√
x2+90000
x2 + 90000
=−2x2 + 180000
(x2 + 90000)32
C′′∣∣x=173.21 > 0
Conclusion? This point does give a local minimum. Therefore, thepoint P where the potatoes should be loaded onto a boat is at 173.21miles due south of point C.
![Page 147: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/147.jpg)
Shipping Example
C′′ =2√
x2 + 90000− 4x2√
x2+90000
x2 + 90000
=−2x2 + 180000
(x2 + 90000)32
C′′∣∣x=173.21 > 0
Conclusion?
This point does give a local minimum. Therefore, thepoint P where the potatoes should be loaded onto a boat is at 173.21miles due south of point C.
![Page 148: 4.6 Applied Optimization - Dr. Travers Page of Mathbtravers.weebly.com/uploads/6/7/2/9/6729909/4.6_applied...Fencing Example Example You want to fence a rectangular region of area](https://reader030.vdocument.in/reader030/viewer/2022040316/5e2a90a4560db0048050c433/html5/thumbnails/148.jpg)
Shipping Example
C′′ =2√
x2 + 90000− 4x2√
x2+90000
x2 + 90000
=−2x2 + 180000
(x2 + 90000)32
C′′∣∣x=173.21 > 0
Conclusion? This point does give a local minimum. Therefore, thepoint P where the potatoes should be loaded onto a boat is at 173.21miles due south of point C.