Question

a rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a single - strand electric fence. with 800 m of wire at your disposal, what is the largest area you can enclose, and what are its dimensions?
the maximum area of the rectangular plot is 80000 m²
the length of the shorter side of the rectangular plot is
the length of the longer side of the rectangular plot is

Explanation:

Step1: Set up variables

Let the length of the side perpendicular to the river be \(x\) (shorter side), and the length parallel to the river be \(y\) (longer side). The total length of the fence is \(2x + y=800\), so \(y = 800 - 2x\). The area \(A=xy=x(800 - 2x)=800x-2x^{2}\).

Step2: Find the maximum of the area function

Take the derivative of \(A(x)\) with respect to \(x\). \(A^\prime(x)=\frac{d}{dx}(800x - 2x^{2})=800-4x\). Set \(A^\prime(x) = 0\), then \(800-4x=0\), which gives \(4x=800\), so \(x = 200\).

Step3: Find the value of \(y\)

Substitute \(x = 200\) into \(y=800 - 2x\). \(y=800-2\times200=400\).

Answer:

The length of the shorter side of the rectangular plot is \(200\space m\).
The length of the longer side of the rectangular plot is \(400\space m\).