Q 8 - Precalculus Enhanced with Graphing Utilities

Question

Beth has 3000 feet of fencing available to enclose a rectangular field.

(a) Express the area A of the rectangle as a function of x, where x is the length of the rectangle.

(b) For what value of x is the area largest?

(c) What is the maximum area?

Step-by-Step Solution

Answer

(a) The area A of the rectangle as a function of x is $A\left(x\right)=-x^2+1500x$

(b) For the area to be the largest x should be 750

(c) Maximum area is 562500

Part (a) Step 1. Given information

Beth has 3000 feet of fencing available to enclose a rectangular field 

Part (a) of Step 2. Area A of the rectangle as a function of x .

Beth has 3000 feet of fencing

Let, x be the length and w be the width

Therefore

 $$\begin{gathered} 2 x+2 w=3000 \\ x+w=1500 \\ w=1500-x \end{gathered}$$

The area of the rectangle is $A=xw$

Substitute the value of w in the equation of the area.

$A=x\left(1500-x\right)$

A as a function of x will be.

$A=-x^2+1500x$

Part (b) of Step 1. value of x .

The value of the area is the largest in the vertex. 

$\begin{array}{l}A=-\left(x^2-2·750·x+{750}^2-{750}^2\right)\\ =-\left(x^2-2·750·x+{750}^2\right)+{750}^2\\ =-{\left(x-750\right)}^2+{750}^2\end{array}$

So, the area is the largest when $x=750$.

Part (c) of Step 1. Maximum area

Substitute x=750 in the equation $A=-x^2+1500x$

$$\begin{gathered} A\left(750\right)=-{750}^2+1500·750 \\ A\left(750\right)=562500 \end{gathered}$$