Im stuck on this question..

Question

A farmer wishes to build three identical rectangular pens side-by-side. The total area of all three pens must be 1500 square feet. What is the least amount of fence which can be used to construct this enclosure? Also, give the domain of the function you're minimizing. Thank you in advance. 

Answer 1

Hey Mike.

 

If the farmer wants 3 identical pens, that means they'll all have the same area. Since the sum of all 3 is 1500 square feet, that tells us each pen will be 500 square feet. So instead of optimizing the amount of fence required for 3 pens of a total of 1500 square feet, let's just optimize 1 at 500 square feet.

 

The area of a rectangle is its length times its width.

 

A = L*W

 

The amount of fence required to build a rectangular enclosure is just the perimeter of the fence.

 

P = 2*L + 2*W

 

We know the area is 500 square feet.

 

500 = L*W

 

So W = 500/L

 

Substitute this into your perimeter equation.

 

P = 2*L + 2*500/L

 

The perimeter is what you want to optimize. So let's take its derivative with respect to L and set it equal to 0.

 

P' = 2 + 1000*(-1/L^2)

 

1000/L^2 = 2

 

2*L^2 = 1000

 

L = sqrt(500) = 5*sqrt(10)

 

Since W = 500/L

 

W = 500/ sqrt(500) = sqrt(500) = 5*sqrt(10)

 

This makes sense that the length and width would be equal, as a square is the optimal shape for a rectangular enclosure.

 

The question didn't ask for what the length and width of each enclosure would be though. It wanted to know the least amount of fence required to make 3 of these. Since they're side by side, the middle enclosure will not need its side lengths taken into account.

 

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There are only 10 lengths of fence required to build the 3 enclosures side by side.

 

10 * W = 50*sqrt(10) feet of fence.

 

Hope this helps.