Chapter 14.7, Problem 14PPS

(a)

To determine

To calculate: Theexpression in factored form of the polynomial expression that represent the area of the walkway.

Answer to Problem 14PPS

The expression in the factored form of the area of the walkway is $12\left(x+3\right)\text{square}\text{meter}$ .

Explanation of Solution

Given Information:

The sides of the squared base $=x\text{meter}$ .

Width of the walkway $=3\text{meter}$ .

Calculation:

The given sides of the squared base is $x\text{meter}$ .

Find the area of the base.

  $\begin{array}{c}\text{Area}\text{of}\text{the}\text{base}=x\times x\\ =x^2\text{square}\text{meter}\end{array}$

Now, find the length and width of the total rectangular area.

  $\begin{array}{c}\text{Length}=3+x+3\\ =x+6\end{array}$

  $\begin{array}{c}\text{Width}=3+x+3\\ =x+6\end{array}$

So, find the total area.

  $\begin{array}{c}\text{Total}\text{area}=\text{Length}\times \text{Width}\\ \text{=}\left(x+6\right)\left(x+6\right)\\ =x\times x+x\times 6+6\times x+6\times 6\\ =x^2+6x+6x+36\\ =x^2+12x+36\end{array}$

Now, find the area of the walkway. Subtract the area of the garden from the total area.

  $\begin{array}{c}\text{Area}\text{of}\text{the}\text{walkway}=\text{Total}\text{area}-\text{Area}\text{of}\text{the}\text{garden}\\ =x^2+12x+36-x^2\\ =12x+36\end{array}$

Since, the greatest common factor of the polynomial $12x+36$ is 12. Thus, convert $12x+36$ into the factored form.

  $\begin{array}{c}12x+36=12\times x+12\times 3\\ =12\left(x+3\right)\end{array}$

Hence, the expression in the factored form of the area of the walkway is $12\left(x+3\right)\text{square}\text{meter}$ .

(b)

To determine

To calculate: The area of the walkway when the base of the statue is 3 meter.

Answer to Problem 14PPS

The area is 108 square meter.

Explanation of Solution

Given Information:

Area of the walkway= $12\left(x+3\right)\text{square}\text{meter}$

Base of the statue=3 meter

Calculation:

The given area of the walkway is $12\left(x+3\right)\text{square}\text{meter}$ .

To find the area, substitute 3 for $x$ in the expression $12\left(x+3\right)$ .

  $\begin{array}{c}\text{Area}=12\left(x+3\right)\\ =12\left(3+3\right)\\ =12\times 9\\ =108\end{array}$

Hence, the area is 108 square meter.