Chapter 9.6, Problem 31PPS

(a)

To determine

To find: The surface area and the volume of the prism.

Answer to Problem 31PPS

The surface area of the prism is $88{\text{cm}}^2$ and the volume of the prism is $48{\text{cm}}^3$ .

Explanation of Solution

The surface area of the prism is calculated as,

  $\begin{array}{c}S=2\left(lb+bh+lh\right)\\ =2\left(\left(6\right)\left(2\right)+\left(2\right)\left(4\right)+\left(6\right)\left(4\right)\right)\\ =2\left(44\right)\\ =88{\text{cm}}^2\end{array}$

The volume of the prism is calculated as,

  $\begin{array}{c}V=lb\\ =6\cdot 2\cdot 4\\ =48{\text{cm}}^3\end{array}$

Therefore, the surface area of the prism is $88{\text{cm}}^2$ and the volume of the prism is $48{\text{cm}}^3$ .

(b)

To determine

To find: The surface area and the volume of the prism for dilation of scale factor of $2$ .

Answer to Problem 31PPS

The surface area of the prism after dilation is $352{\text{cm}}^2$ and the volume of the prism after dilation is $384{\text{cm}}^3$ .

Explanation of Solution

The new dimension after dilation is,

  $\begin{array}{c}l=6\left(2\right)=12\\ b=2\left(2\right)=4\\ h=4\left(2\right)=8\end{array}$

The surface area of the prism is calculated as,

  $\begin{array}{c}S=2\left(lb+bh+lh\right)\\ =2\left(\left(12\right)\left(4\right)+\left(4\right)\left(8\right)+\left(12\right)\left(8\right)\right)\\ =2\left(176\right)\\ =352{\text{cm}}^2\end{array}$

The volume of the prism is calculated as,

  $\begin{array}{c}V=lb\\ =12\cdot 4\cdot 8\\ =384{\text{cm}}^3\end{array}$

Therefore, the surface area of the prism after dilation is $352{\text{cm}}^2$ and the volume of the prism after dilation is $384{\text{cm}}^3$ .

(c)

To determine

To find: The surface area and the volume of the prism for dilation of scale factor of $\frac{1}{2}$ .

Answer to Problem 31PPS

The surface area of the prism after dilation is $22{\text{cm}}^2$ and the volume of the prism after dilation is $6{\text{cm}}^3$ .

Explanation of Solution

The new dimension after dilation is,

  $\begin{array}{c}l=6\left(0.5\right)=3\\ b=2\left(0.5\right)=1\\ h=4\left(0.5\right)=2\end{array}$

The surface area of the prism is calculated as,

  $\begin{array}{c}S=2\left(lb+bh+lh\right)\\ =2\left(\left(3\right)\left(1\right)+\left(1\right)\left(2\right)+\left(3\right)\left(2\right)\right)\\ =2\left(11\right)\\ =22{\text{cm}}^2\end{array}$

The volume of the prism is calculated as,

  $\begin{array}{l}V=lb\\ =3\cdot 1\cdot 2\\ =6{\text{cm}}^3\end{array}$

Therefore, the surface area of the prism after dilation is $22{\text{cm}}^2$ and the volume of the prism after dilation is $6{\text{cm}}^3$ .

(d)

To determine

To find: The number of times the surface area and volume of the image is after each dilation.

Answer to Problem 31PPS

The number of times the surface area and volume of the image is after each dilation is mentioned below.

Explanation of Solution

For scale factor 2.

  $\begin{array}{c}\frac{S_2}{S_0}=\frac{352}{88}\\ =4\\ S_2=4S_0\end{array}$

For scale factor $\frac{1}{2}$ .

  $\begin{array}{c}\frac{S_{1/2}}{S_2}=\frac{22}{88}\\ S_{1/2}=\frac{1}{4}{S}_2\\ =\frac{1}{4}\end{array}$

For scale factor 2.

  $\begin{array}{c}\frac{V_2}{V_0}=\frac{384}{48}\\ =8\\ V_2=8V_0\end{array}$

For scale factor $\frac{1}{2}$ .

  $\begin{array}{c}\frac{V_{1/2}}{V_2}=\frac{6}{48}\\ V_{1/2}=\frac{1}{8}{V}_2\end{array}$

(e)

To determine

To make: The conjecture on surface area and volume of prism due to dilation.

Answer to Problem 31PPS

The conjecture is stated below.

Explanation of Solution

From the obtained result.

Surface area will have effect by $r^2$ on dilation.

Volume will have effect by $r^3$ on dilation.