Chapter 3.4, Problem 8AYU

(a)

To determine

To express: The area $A$ of the rectangle as a function of x .

Answer to Problem 8AYU

  $A=x\left(1500-x\right)$

Explanation of Solution

Given: Beth has 3000 feet of fencing available to close a rectangular filed.

Let width of rectangular filed be $y\text{ ft}$ and length be $x\text{ ft}$ .

Fencing of the rectangular filed is 3000 ft.

So, $x+y+x+y=3000$

  $y=1500-x$

Area of rectangle, $A=xy=x\left(1500-x\right)$

(b)

To determine

To find: The value of x when area of rectangular filed maximum area.

Answer to Problem 8AYU

  $x=750\text{ ft}$

Explanation of Solution

Given: Beth has 3000 feet of fencing available to close a rectangular filed.

The function of area of rectangular filed, $A=x\left(1500-x\right)$.

Now, find the maximum area of filed.

  $\begin{array}{l}A=1500x-x^2\\ A\text{'}=1500-2x\end{array}$

For maximum or minimum, $A\text{'}=0$

  $\begin{array}{c}1500-2x=0\\ x=750\end{array}$

  $A\text{'}\text{'}=-2<0\left(\max \right)$

So, area of filed is maximum when $x=750\text{ ft}$ .

(c)

To determine

To find: The maximum area of filed.

Answer to Problem 8AYU

  $562500{\text{ ft}}^2$

Explanation of Solution

Given: Beth has 3000 feet of fencing available to close a rectangular filed.

The maximum area of filed at $x=750\text{ ft}$ .

  $A_{\max }=750\left(1500-750\right)=562500{\text{ ft}}^2$