word problem

Question

Nate wants to build a rectangular pen for his animals. One side of the pen will be against the barn; the other three sides will be enclosed with wire fencing. If Nate has 650 feet of fencing, what dimensions would maximize the area of the pen?

a) Let w be the length of the pen perpendicular to the barn. Write an equation to model the area of the pen in terms of w

Area =

b) What width w would maximize the area?

w =

Answer 1

Alli,

 

With these sorts of word problems, I always draw a picture of what the problem describes, so I can be clear conceptually with what's going on. (Also, I got a 5 on the BC calculus exam!) When you draw the picture, how many sides will the fence have?

 

Part a)

 

Now that you have your picture, let's turn to part a). Start with the formula for area, A=w*l. The problem is asking you to put A solely in terms of w, which means you have to substitute l for something else.

 

You can substitute for l from the equation for perimeter. We know from the problem description that the fence is 650 feet long, so perimeter=650 feet. But we can't use the standard formula for perimeter, which is 2l + 2w = 650. Why not?

 

Here'a hint: Think about how many sides the fence has. What formula should we use instead? And once you have that formula, you can solve for l in terms of w.

 

After you get l in terms of w, you can substitute it into A, and you'll have finished part a).

 

Part b)

 

Now you have A in terms of w. You can use calculus techniques to optimize A. Simply take the derivative, set it equal to zero, and find the critical point(s). Next, confirm that the critical point(s) is a maximum. This will give you the w that maximizes area, solving your problem. It's the same as optimizing y = -x².

 

Best,

Bryce