Chapter 7.5, Problem 47E

a.

To determine

To write: An expression that represents the area being paved.

Answer to Problem 47E

Expression is $-x^2+38x$ .

Explanation of Solution

Given information: A figure representing the area is given.

Calculation:

  $\begin{array}{l}\begin{array}{cc}& \begin{array}{ll}a\hfill & \hfill \\ \hfill & \hfill \end{array}\end{array}\\ \begin{array}{llll}b\hfill & c\hfill & \hfill & d\hfill \\ \hfill & \hfill & \hfill & \hfill \end{array}\end{array}$

Width of region $b$ $=(\frac{20-x}{2})$

Length of region $b$ $=x$

Area of region $b$ $=(10-\frac{x}{2})(x)$

  $=10x-\frac{x^2}{2}$

Area of region $d$ $=$ Area of region $b$

  $=10x-\frac{x^2}{2}$

Area of region $c$ $=x^2$

Area of region $a$ $=(x)(18-x)$

Total area $=$ Area of region $a$ +Area of region $b$ +Area of region $c$ +Area of region $d$

Total area $=(x)(18-x)+x^2+2(10x-\frac{x^2}{2})$

  $\begin{array}{l}=18x-x^2+x^2+20x-x^2\\ =-x^2+38x\end{array}$

b.

To determine

To find: Find width when area is $280sq.m$ .

Answer to Problem 47E

The width $x$ is $10m$ .

Explanation of Solution

Given information: Area is $280sq.m$ .

Calculation:

Expression representing area is $-x^2+38x$ (From part a)

  $-x^2+38x=280$

  $-x^2+38x-280=0$

Writing left hand side factored and simplifying,

  $\begin{array}{l}-x^2+28x+10x-280=0\\ x(-x+28)-10(-x+28)=0\\ (x-10)(-x+28)=0\\ x=10,28\end{array}$

As evident from the figure $x<18$ , $\therefore x=10$ .

Thus, the width $x$ is $10m$ .