Chapter 12.8, Problem 26PPS

a.

To determine

To find the height of the larger cylinder.

Answer to Problem 26PPS

  $13.5$ cm

Explanation of Solution

Given:

Two similar cylinders.

The height of the cylinders are in the ratio $2$ to $3$ .

Lateral surface area of the larger cylinder is $162\pi$ square centimeters.

Diameter of the smaller cylinder is $8$ cm.

Concept used:

Two solids are similar if they are the same type of solid and their corresponding radii and heights are proportional.

Formula used:

Lateral surface area of a cylinder $=2\pi rh$ .

Calculation:

Since the two cylinders are similar, and the heights are in the ratio $2$ to $3$ , so find the diameter of the larger cylinder.

  $\frac{2}{3}=\frac{8}{d}$

  $\begin{array}{l}\Rightarrow 2\cdot d=8\cdot 3\\ \Rightarrow 2d=24\\ \Rightarrow d=12\end{array}$

So, the diameter of the larger cylinder is $12$ cm, so the radius is $6$ cm.

Now, using this radius and the lateral surface area to find the height of the larger cylinder.

Lateral surface area of the larger cylinder $=2\pi rh$ .

  $\Rightarrow 162\pi =2\pi \left(6\right)h$

  $\Rightarrow 162\pi =12\pi h$

  $\Rightarrow 162=12h$

  $\Rightarrow 13.5=h$

Conclusion:

Therefore, the height of the larger cylinder is $13.5$ cm.

b.

To determine

To sketch and label the two cylinders.

Explanation of Solution

Given:

Two similar cylinders.

The height of the cylinders are in the ratio $2$ to $3$ .

Lateral surface area of the larger cylinder is $162\pi$ square centimeters.

Diameter of the smaller cylinder is $8$ cm.

Diameter of large cylinder is $12$ cm.

Height of large cylinder is $13.5$ cm.

Height of small cylinder is $9$ cm.

Concept used:

Two solids are similar if they are the same type of solid and their corresponding radii and heights are proportional.

Calculation:

  

c.

To determine

To find the ratio of volumes of both cylinders.

Answer to Problem 26PPS

  $3.375$ times

Explanation of Solution

Given:

Two similar cylinders.

The height of the cylinders are in the ratio $2$ to $3$ .

Lateral surface area of the larger cylinder is $162\pi$ square centimeters.

Diameter of the smaller cylinder is $8$ cm.

Formula used:

Volume of a cylinder $=\pi r^{2}h$

Calculation:

First find the height of the smaller cylinder.

Since the two cylinders are similar, and the heights are in the ratio $2$ to $3$ , so the height of the smaller cylinder,

  $\frac{2}{3}=\frac{h}{13.5}$

  $\begin{array}{l}\Rightarrow 2\cdot 13.5=h\cdot 3\\ \Rightarrow 27=3h\\ \Rightarrow 9=h\end{array}$

Simplify the ratio of the volume of the larger cylinder to the volume of the smaller cylinder.

  $\frac{V_{\text{large cylinder}}}{V_{\text{small cylinder}}}=\frac{\pi {\left(6\right)}^2\left(13.5\right)}{\pi {\left(4\right)}^2\left(9\right)}$

  $=\frac{486}{144}$

  $=3.375$

  $\Rightarrow V_{\text{large cylinder}}=3.375\cdot V_{\text{small cylinder}}$

Conclusion:

Therefore, the volume of the larger cylinder is $3.375$ times as great as the volume of the smaller cylinder.