Question

A farmer wishes to enclose a pasture that is bordered on one side by a river(so one of the four sides doesn't require fencing). She has decided to create a rectangular shape for the area and will use barbed wire to create the enclosure. There are 600 feet of wire available for the project, and she will use all the wire. What is the maximum area that can be enclosed by the fence?

Answer 1

this may help you solve the problem

Answer 2

you can find an equation to solve length l or width w using perimeter such as \[600=l+2w\]\[l=600-2w~~~~~(1)\]using area of a rectangle formula\[A=lw~~~~~(2)\]substitute (1) to (2) and apply completing the square\[A=(600-2w)w\]\[A=600w-2w^2\]or\[-\frac{A}{2}=w^2-300w\]\[-\frac{A}{2}+22500=w^2-300w+22500\]\[-\frac{A}{2}+22500=(w-150)^2\]the width of the lot is\[w-150=0\]\[w=150~~feet\]and the maximum area enclosed by the 600 feet fence is\[-\frac{A}{2}+22500=0\]\[\frac{A}{2}=22500\]\[A=45000~~ft^2\]

Answer 3

Another solution is by means of Differential Calculus, applying maxima and minima... \(\ddot\smile\)