Question

Enclosing the Most Area with a Fence: A farmer with 4000 meters of fencing wants to enclose a rectangular plot that borders on a river. If the farmer does not fence the side along the river, what is the largest area that can be enclosed? ( See the figure)

Short answer

The maximum are enclosed here is$2,000,000$square meters.

Step-by-step solution

Step 1. Given information.

Given that a farmer with 4000 meters of fencing wants to enclose a rectangular plot that borders on a river and the farmer does not fence the side along the river,

Step 2. Express the area A as a function of x (length of the rectangle).

Let the length of the rectangular field be x and

the width of the rectangle is $4000-2x$.

Because $$\begin{gathered} 2l+w=4000 \\ 2x+w=4000 \\ w=4000-2x \end{gathered}$$

The area of a rectangle is $A=lw$.

We get

$$\begin{gathered} A=x(4000-2x) \\ =4000x-2x^2 \\ =-2x^2+4000x \end{gathered}$$

Step 3. Find the maximum area enclosed.

The function A is a quadratic function with $a=-2,b=4000$, and $c=0$. As $a<0$, the vertex is the highest point on the parabola.

The area A is maximum when x is

$$\begin{gathered} x=-\frac{b}{2a} \\ =\frac{-4000}{2(-2)} \\ =1000 \end{gathered}$$

Substitute $x=1000$in $A=-2x^2+4000x$.

$$\begin{gathered} A=-2{\left(1000\right)}^2+4000\left(1000\right) \\ =-2000000+4000000 \\ =2,000,000 \end{gathered}$$

The maximum area enclosed is 2,000,000 square meters.