Calculus word problem

Question

A farmer wants to enclose a 192 square foot rectangular area with a fence that cost $3 per foot, and subdivide it into two equal-area rectangular pens with fence that costs $2 per foot. If the farmer wants to minimize the total cost of the fence, then the total cost of the fence will be

Answer 1

Hey Zachary.

 

We don't know much about the exact dimensions of this 192 ft² rectangular area, but we know enough to solve the problem.

 

The area of a rectangle is:

 

A = L*W

 

We know the area our fence needs to enclose is 192, so:

 

192 = L*W

 

The perimeter of this area is going to be 2L + 2W. This is how much $3 fence will be required.

 

If he wants to subdivide this area into two equal parts, he'll have to add another fence of length W, which costs him $2 per foot.

 

He wants to minimize cost. So we need to figure out a cost function.

 

C(L,W) = 3(2L + 2W) + 2*W

 

This tells us that we will spend $3 per foot on the perimeter of fencing, and $2 per foot on the extra fencing down the middle.

 

C(L,W) = 6L + 6W + 2W = 6L + 8W

 

Now, we also know the L*W = 192, meaning we can say that

 

L = 192/W

 

So

 

C(W) = 6*(192/W) + 8W

 

C(W) = 1152/W + 8W

 

If we want to minimize this, we'll have to take the derivative and set it equal to zero.

 

C' = -1152/W² + 8

 

0 = -1152/W² + 8

 

1152/W² = 8

 

8W² = 1152

 

W² = 144

 

W = 12 or -12, but since width can't be negative, it will just be 12.

 

If W = 12, then

 

L = 192/12 = 16

 

So

 

C(L,W) = 6L + 8W

 

C(16,12) = 6(16) + 8(12) = 96 + 96 = 192

 

The farmer is going to spend at least $192 on this project.

 

Hope this helps.