Question

http://openstudy.com/groups/mathematics/updates/4e326a4a0b8ba7b2da416ae8

Answer 1

Your question is slightly different from the normal way these are asked, as one side is a Barn.

You have a rectangular garden plot, where one side of the rectangle is the Barn and the reminding three sides are constructed from the 100 ft of fencing that the farmer has. Therefore if we let the side of the rectangle opposite the barn be of length \(y\) and the other two sides be each of length \(x\), you have that \[2x+y=100\] and the area of the plot is \[A=x y=x(100-2x)=100x -2x^2\] where we have used the \(2x+y=100\) and rearranged it to find \(y=100-2x\)

Now you want the area to be maximum. So differentiate \(A\) with respect to \(x\)
\[\frac{dA}{dx}=100-4x\] and set equal to zero \[100-4x=0 \textrm{ when } x=25\] now fill this back in to find \(y\) \[y=100-2x=100-2\times 25=50\]
So your answer is that the Area of the plot is \[A=xy=25 \times 50 = 1250\] and it has dimensions \[x=25, y=50\]