this is problem 49 page 219:
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This is problem 49 page 219: - PowerPoint PPT PresentationTRANSCRIPT
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This is problem 49 page 219:
A man is in a boat 2 miles from the nearest point on the coast. He is to go to a point Q, located 3 miles down the coast and 1 mile inland. If he can row at 2 miles per hour and walk at 4 miles per hour, toward what point on the coast should he row in order to reach point Q in the least amount of time.
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Need a picture:
Q
1 mile
2 miles
3 miles
x
3 - x
Find x to minimize time
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Use the Pythagorean theorem twice to find the distance to row and walk
To row: 42 x
To walk: 1)3( 2 x
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The time to row is: milesxmiles
hour4
22
Notice I used unit analysis to find I had to divide by 2 and not multiply by 2
The time to walk is:
milesxmiles
hour1)3(
42
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The primary equation is the total time:
1)3(4
14
2
1 22 xxT
We don’t need a “secondary equation”. Why?
The primary equation only has one variable, there is no need to substitute for a second variable.
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1)3(4
14
2
1 22 xxT
To minimize, find the derivative = 0
This will be good chain rule practice
))1)(3(2(1)3(
2
14124
2
121' 2
122
12 xxxxT
Outside Inside Outside Inside
Do you see another chain rule here?
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))1)(3(2(1)3(
2
14124
2
121' 2
122
12 xxxxT
We need to solve this mess = 0
First, let’s clean it up a bit by multiplying constants
)3(1)3(4
1421' 2
122
12 xxxxT
Wow, let’s cheat and use MathPert to graph this and find the solution
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Click on this
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Now this
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Select “Type it in”
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note the use of ( ) everywhere and the use of negative exponents now
click OK
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Make sure this is the equation we want and it is. Click OK again
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Here is our graph, we want where it equals 0, ie, the x-intercept
click this about 4 times to get a better view
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This looks like the point
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Use the Point and Slope Tool
click it and center the crosshairs on the intercept
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y = 0 and x = 1
this is where the derivative is equal to 0!
We just use a ‘numerical” technique to solve the equation
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Just to make sure, evaluate at x = 1:
)3(1)3(4
1421' 2
122
12 xxxxT
You should get:
054
2
52
1
The man should aim for a point 1 mile down the coast
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)3(1)3(4
1421' 2
122
12 xxxxT
To review what I typed into MathPert:
1/2((x^2+4)^(-1/2)(x))-1/4((3-x)^2+1)(-1/2)(3-x))
“Count” the number of opening (( versus the number of closing )), they must match in any expression. I could also type, which is easier:
x/2sqrt(x^2+4)-(3-x)/4sqrt((3-x)^2+1)