mathematical models constructing functions. suppose a farmer has 50 feet of fencing to build a...

Mathematical M

odels

Constructing Functions

Suppose a farmer has 50 feet of fencing to build a rectangular corral. Express the rectangular area A he can enclose as a function of the length x of a side. Then find the dimensions to make his corral to enclose the maximum area.

Draw a picture x

w

Total fencing needed would be the perimeter (adding up all sides)

5022 wxx

w

xwAArea of rectangle is length x times width w

This is the area as a function of x and w. We want area as a function of x.


x

w

If we solve for w in this equation, we can substitute it in for w in the area equation below.

5022 wxx

w

xwAx

xw

25

2

250

x25


To find maximum area, we’ll look at the graph.


x

w

x

w

xxA 25


The graph is a parabola that opens down. Put this in a graphing calculator and trace the x where f(x) is at its maximum.

Adjust the window until you get a good view.

This is on the next screen.

Remember x is the side of the rectangle and f(x) is the area.

This would be the x value that would

give the maximum area

This would be the maximum

area.

(12.5, 156.25)

The maximum enclosed area would be 156.25 square feet

Another Example

Let P = (x, y) be a point on the graph of y = x2 – 8

a) Express the distance d from P to the point (0, -1) as a function of x.

b) What is d if x = 0?

c) What is d if x = -1?

d) Use a graphing utility to graph d = d(x).

e) For what values of x is d smallest?

The first thing to do is draw a picture. We’ll take each part and do it on a slide.



This is a parabola vertically shifted down 8.

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

(x, y)

(0, -1)



Let’s use the distance formula to

express the distance from (x, y) to (0, -1)

2122

12 yyxxd

22 10 yxd

This is a formula for the distance from P to (0, -1) as a function of x and y. We only want it as a function of x so we need another equation relating x and y to solve and substitute for y.

22 1 yxd

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

(x, y)

(0, -1)



Since P is a point on the graph of y = x2 – 8, this equation will be true about the relationship between x and y

22 1 yxd

We can then substitute for y in the distance equation above. y = x2 – 8

222 18 xxd

222 7 xx

b) What is d if x = 0?

222 7 xxdSo we have our formula for the distance from P to (0, -1) and we are ready to answer other parts of the question.

222 7 xxd 222 700 d 749

c) What is d if x = -1?

222 711 d 37361

d) Use a graphing utility to graph d = d(x).

e) For what values of x is d smallest?

222 7 xxd

This is an even function so will also be smallest d at x = - 2.55

Two cars are approaching an intersection. One is 2 miles south of the intersection and is moving at a constant speed of 30 miles per hour. At the same time, the other car is 3 miles east of the intersection and is moving at a constant speed of 40 miles per hour.

Express the distance d between the cars as a function of time.

Let’s draw a picture putting the cars on a coordinate system letting the origin be the intersection.

(0, -2)

(3, 0)

The first car is moving along the y axis so its position at any time is changing but can be written as (0, y)

The second car is moving along the x axis so its position at any time is changing but can be written as (x, 0)

Using the distance formula, we can find the distance between (x, 0) and (0, y) to find the distance between the two cars

22 00 yxd

Two cars are approaching an intersection. One is 2 miles south of the intersection and is moving at a constant speed of 30 miles per hour. At the same time, the other car is 3 miles east of the intersection and is moving at a constant speed of 40 miles per hour.

Express the distance d between the cars as a function of time.

We need to find equations for x and y in terms of t

22 yxd

We now have the distance as a function of time

22 302403 ttd

The first car is moving along the y axis. Using d = rt we have d = 30t. It started at -2 on the y axis so it’s y axis position is y = -2 + 30t

Similarly the second car is moving along the x axis. Using d = rt we have d = 40t. It started at 3 on the x axis but is moving in the negative x direction so it’s x axis position is x = 3 – 40t

22 302403 ttd

By looking at the graph of the distance between the two cars, determine if the cars crash at the intersection and if not, find the minimum distance between them.

Here is a graph showing t on the x axis and the distance d on the y axis. Looks like the distance gets close to 0 so let’s zoom in and see if it ever is (meaning the cars did crash).

They don’t crash and the closest they get is about ¼ mile apart.

mathematical models constructing functions. suppose a farmer has 50 feet of fencing to build a...

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