Chapter 6.2, Problem 42PPS

a.

To determine

To find: The representation of the situation in the algebra titles and the width of the rectangle.

Answer to Problem 42PPS

The width of the rectangle is $b=x+3$ .

Explanation of Solution

Given information:

The area of the rectangle is $2x^2+7x+3$ and the length of the rectangle is $2x+1$ .

The representation of the situation in the algebra titles is written as, $\begin{array}{l}A=l\times b\\ b=\frac{A}{l}\\ b=\frac{2x^2+7x+3}{2x+3}\end{array}$

The width of the rectangle is calculated as,

  $\begin{array}{c}b=\frac{2x^2+7x+3}{2x+1}\\ =\frac{2x^2+6x+x+3}{2x+1}\\ =\frac{2x\left(x+3\right)+1\left(x+3\right)}{2x+1}\\ =\frac{\left(2x+1\right)\left(x+3\right)}{2x+1}\end{array}$

Simplify further,

  $b=x+3$

Thus, the width of the rectangle is $b=x+3$ .

b.

To determine

To write: The expression to represent the model.

Answer to Problem 42PPS

The expression to represent the model is $b=x+3$ .

Explanation of Solution

Given information:

The area of the rectangle is $2x^2+7x+3$ and the length of the rectangle is $2x+1$ .

The expression to represent the model is written as,

  $\begin{array}{l}A=l\times b\\ b=\frac{A}{l}\\ b=\frac{2x^2+7x+3}{2x+3}\end{array}$

Thus, the width of the rectangle is $\frac{2x^2+7x+3}{2x+3}$ .

c.

To determine

To find: The representation of the situation in the algebra titles and the width of the rectangle.

Answer to Problem 42PPS

The width of the rectangle is $b=x+3$ .

Explanation of Solution

Given information:

The area of the rectangle is

  $2x^2+7x+3$ and the length of the rectangle is $2x+1$ .

The representation of the situation in the algebra titles is written as,

  $\begin{array}{l}A=l\times b\\ b=\frac{A}{l}\\ b=\frac{2x^2+7x+3}{2x+3}\end{array}$

The width of the rectangle is calculated as,

  $\begin{array}{c}b=\frac{2x^2+7x+3}{2x+1}\\ =2x+1\begin{array}{c}\hfill x+3\\ \hfill \overline{)\begin{array}{l}2x^2+7x+3\\ \underset{\_}{2x^2+x}\\ 6x+3\\ \underset{\_}{6x+3}\\ 0\end{array}}\end{array}\\ b=x+3\end{array}$

Thus, the width of the rectangle is $b=x+3$ .

The solution for the concrete is same as algebraic model.