University Level Calculus

Question

You are asked to design a rectangular barnyard enclosing 20,000 square feet with fencing on three sides and the wall of a long barn on the fourth. Find the shortest length of fence needed.

 

I am just starting my summer semester in Calc I and just need a little push in the right direction.

Answer 1

Let w be the width of the barnyard and l be the length.  The area is wl, and that's equal to 20000.

 

Let's assume that the barn takes up a side along the length, so the perimeter is l + 2w.  It's the perimeter that you want to minimize.

 

Since wl = 20000, l = 20000 / w.

 

That means that the perimeter is 20000 / w + 2w = 20000 w^-1 + 2w.

 

Differentiate that with respect to w...

 

D_x 20000 w^-1 + 2w = -20000 w^-2 + 2 = -20000 / w² + 2

 

Set that equal to 0 and solve:

-20000 / w² + 2 = 0

2 = 20000 / w²

2 / 20000 = w²

1 / 10000 = w²

±100 = w

 

Since the width can't be negative, you can eliminate the negative solution.  Therefore w = 100

 

Since wl = 20000, l must be 200.

 

Therefore the minimum perimeter is 200 + 2 * 100 = 400 feet.