Question

A farmer wants to construct a rectangular shaped fenced-in area on his property. To do so, he plans on using a river that runs in a straight line along the back of his property as one of the side of his rectangle. He will use the 300 feet of fence he has to build two sides perpendicular to the river and one side parallel to the river what is the largest area of his yard he convention? Write your equation and answer the question that is asked

Answer 1

check the first picture below.

bearing in mind that, the river side doesn't get any fencing, because the river itself is acting as a fence, then the 300 feet will be only for those two widths there and the length.


`\bf A=width\cdot length\implies A(w)=w~\stackrel{length}{(~300-2w)} \\\\\\ A(w)=300w-2w^2\implies \cfrac{dA}{dw}=300-4w\implies 0=300-4w \\\\\\ 4w=300\implies w=\cfrac{300}{4}\implies w=75`

now, doing a first-derivative test on the regions before and after the critical point of 75, check the second picture below, 75 is clearly a maximum.

so, the largest Areas is at A(75) = 11250.