Chapter 3.4, Problem 9AYU

Enclosing a Rectangular Field 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.)

To determine

To calculate: The largest area of the fence that can be enclosed leaving the side that borders the river.

Answer to Problem 9AYU

Solution:

The maximum area of the rectangle is $2,000,000m^2$.

Explanation of Solution

Given:

The farmer is having a fence of 4000 meters.

Formula used:

Perimeter of a rectangle is twice the sum of its length and width.

Area of a rectangle is $A=lw$, where l is the length and w is the width.

Calculation:

Let the length of the rectangle be l and width be x.

The 4000 meters of fencing can be considered as the perimeter of the rectangle.

One side of the fence that borders the river need not be enclosed; therefore, the perimeter will be the sum of 2 sides of width and one side of length.

Therefore, we get

$l+2x=4000$

$\Rightarrow l=4000-2x$

Now, we have to find the area of the rectangle.

Thus, we get the area as

$A=(4000-2x)x=4000x-2x^2$

The equation of area is a quadratic equation with $a=-2,b=4000,c=0$. Since $a$ is negative, the vertex is the maximum point of the equation.

Thus, we get the maximum point at $-\frac{b}{2a}=-\frac{4000}{2(-2)}=1000$

Therefore, the area is maximum when the length is 1000 meters.

The maximum area of the rectangle is

$A\left(-\frac{b}{2a}\right)=A(1000)=4000(1000)-(1000{)}^2$

$=2000000$

The maximum area of the rectangle is $2,000,000m^2$.