Chapter 6.8, Problem 1E

To determine

To calculate: The maximum amount of area encloses by him,

A farmer has $240$ feet of fencing that is used by him to enclose a rectangular area for his chickens.

Answer to Problem 1E

The maximum area encloses by him is: $7200{\text{ ft}}^2$ , option A is right answer.

Explanation of Solution

Given information:

A farmer has $240$ feet of fencing that is used by him to enclose a rectangular area for his chickens.

As he planes to build the enclosure using the wall of his barn as one of the walls.

Here the options are:

  $\begin{array}{l}A7200{\text{ ft}}^2\\ B{\text{ 4960 ft}}^2\\ C{\text{ 3600 ft}}^2\\ D{\text{ 3280 ft}}^2\end{array}$

Formula used:

Area of rectangle: $l\times b$ ,

Perimeter of rectangle:

  $A=2b+2l$

Here,

  $\begin{array}{l}l=\text{length}\\ b=\text{breadth}\end{array}$ .

Calculation:

Consider the information,

A farmer has $240$ feet of fencing that is used by him to enclose a rectangular area for his chickens.

As he planes to build the enclosure using the wall of his barn as one of the walls.

Recall that Perimeter of rectangle:

  $A=2b+2l$

Here perimeter is: $240$

Apply the value in the formula of perimeter:

  $\begin{array}{c}240=2b+l\\ 240-2b=l\end{array}$

Rewrite the equation as:

  $\begin{array}{c}240-2b=l\\ l=240-2b\end{array}$

Recall that Area of rectangle: $l\times b$ ,

Apply the value of $l$ in the formula of area:

As,

  $\begin{array}{l}A=l\times b\\ A=\left(240-2b\right)\cdot \left(b\right)\\ A=b\cdot \left(240-2b\right)\\ A=240b-2b^2\end{array}$

Rewrite the expression as:

  $\begin{array}{l}A=240b-2b^2\\ A=-2b^2+240b\end{array}$

Write this equation $A=-2b^2+240b$ in the form of $x$ .

  $\begin{array}{l}A=-2b^2+240b\\ A=-2x^2+240x\end{array}$

Here to solve the equation apply the formula: $x=\frac{-b}{2a}$ .

Apply the values in the formula $x=\frac{-b}{2a}$ from the equation $A=-2x^2+240x$ .

As, $b=240,a=-2$

  $\begin{array}{c}x=\frac{-b}{2a}\\ x=\frac{-240}{2\left(-2\right)}\\ x=\frac{-240}{-4}\\ x=60\end{array}$

Here,

  $\begin{array}{c}f\left(x\right)=-2x^2+240x\\ x=60\\ f\left(60\right)=-2{\left(60\right)}^2+240\left(60\right)\\ f\left(60\right)=-7200+14400\end{array}$

Simplified further as:

  $\begin{array}{l}f\left(60\right)=-7200+14400\\ f\left(60\right)=7200\end{array}$

The maximum area is: $7200{\text{ ft}}^2$

Thus, the maximum area encloses by him is: $7200{\text{ ft}}^2$ , option A is right answer.