SOLUTION: A rectangular garden with an area of 2050 square feet is to be located next to a barn with fencing on three sides and the barn acting as the fourth side. Write a function which giv

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Question 112897: A rectangular garden with an area of 2050 square feet is to be located next to a barn with fencing on three sides and the barn acting as the fourth side. Write a function which gives the length of fencing needed if the length of the side parellel to the barn is x. Find the dimensions of the garden using the minimum amount of fence.
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!
If the side parallel to the barn is x and the area is 2050, then the two sides perpendicular to the barn are each 2050%2Fx.

Let L(x) be the length function and L%28x%29+=+x.
Let W(x) be the width function and W%28x%29=2050%2Fx
Let P(x) be the perimeter function representing the length of the three fenced sides, hence P%28x%29=L%28x%29%2B2W%28x%29

P(x) will have a critical point where dP%28x%29%2Fdx+=+0.

dP%28x%29%2Fdx=dL%28x%29%2Fdx%2B2dW%28x%29%2Fdx

dP%28x%29%2Fdx=1-2%2A2050%2Fx%5E2

1-2%2A2050%2Fx%5E2=0
x%5E2-4100=0
x=sqrt%284100%29

Now the question is whether this critical point is a minimum or a maximum.

If d%5E2P%28sqrt%284100%29%29%2Fdx%5E2%3E0 then the critical point is a minimum.

d%5E2P%28x%29%2Fdx%5E2=2%2A4100%2Fx%5E3

d%5E2P%28sqrt%284100%29%29%2Fdx%5E2=2%2A4100%2F%28sqrt%284100%29%29%5E3%3E0, therefore the critical point is a minimum.

So sqrt%284100%29 is the length x that gives the minimum length fence. The sides are then 2050%2Fsqrt%284100%29 or sqrt%284100%29%2F2 when you rationalize the denominator. I would submit the answer in this form since these values are exact. If you want to do the calculator work, you will get x to be a little larger than 64 (since 64%5E2=4096 and the sides will be a little more than 32, giving you a bit more than 128 feet of fencing required.

Hope this helps,
John