Q. 10 - Precalculus Enhanced with Graphing Utilities

Question

A farmer with $2000$ m of fencing wants to enclose a rectangular plot that borders on a straight highway. If the farmer does not fence the side along the highway, what is the largest area that can be enclosed?

Step-by-Step Solution

Answer

The largest area that can be enclosed is $250,000m^2$.

Step 1. Given information.

Consider the given question,

Perimeter of the rectangular plot$=2000m$

Assume the length and breadth of the plot to be l and w.

We know, perimeter of rectangle, $P=2\left(l+b\right) ...... \left(i\right)$

Substitute the values in equation (i),

$$\begin{gathered} 2000=2\left(l+w\right) \\ l=1000-w \end{gathered}$$

We know, area of rectangle, $A=l\times b ...... \left(ii\right)$

Substitute the values in equation (ii),

$$\begin{gathered} A=\left(1000-w\right)w \\ A=1000w-w^2 \end{gathered}$$

So, we can say the area as a function of w is $A\left(w\right)=-w^2+1000w$.

Step 2. Compare the function with the general quadratic equation.

On comparing the function with the general quadratic equation, we get,

$$\begin{gathered} a=-1, \\ b=100, \end{gathered}$$

As $a<0$, the vertex is the highest point on the parabola that represents the function. The area is thus maximum when the value of $l=-\frac{b}{2a} ...... \left(iii\right)$

Substitute the values in equation (iii),

$$\begin{gathered} x=-\frac{1000}{2\left(-1\right)} \\ x=500 \end{gathered}$$

Therefore, the area is largest when the width is $500m$.

Step 3. Find the length of the plot.

Substitute the values in the equation of l,

$$\begin{gathered} l=1000-500 \\ l=500m \end{gathered}$$

Substitute the values of l and b in equation (ii),

$$\begin{gathered} A=500\times 500 \\ A=250,000m^2 \end{gathered}$$