Chapter 12.8, Problem 21PPS

To determine

To find the volume of the larger solid.

Answer to Problem 21PPS

  $2439.6$ cubic centimeters

Explanation of Solution

Given:

Two similar solids

  

Concept used:

If the two solids are similar, then the ratio of the volumes of the larger and smaller solids equals the cube of the ratio of the diameters of their circular bases.

Formula used:

Volume of a cone $=\frac{1}{3}\pi r^{2}h$

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

Calculation:

For the given two solids, the diameter of the base of the smaller solid is $2\times 6$ or $12$ cm.

So, the ratio of the diameters is $\frac{12}{16}=\frac{3}{4}$ .

First find the volume of the smaller solid.

Each solid is a composite composed of a cone on top of a cylinder.

The volume of the smaller solid will equal the volume of the cone with a radius of $6$ cm and a height of $5.7$ cm and the volume of a cylinder with a radius of $6$ cm and a height of $7.2$ cm.

Volume of smaller solid $=$ volume of the cone $+$ volume of the cylinder.

  $V_{\text{smaller solid}}=V_{\text{cone}}+V_{\text{cylinder}}$

  $=\frac{1}{3}\pi r^{2}h+\pi r^{2}h$

  $\begin{array}{l}=\frac{1}{3}\pi {\left(6\right)}^2\left(5.7\right)+\pi {\left(6\right)}^2\left(7.2\right)\\ =\frac{1}{3}\pi \left(36\right)\left(5.7\right)+\pi \left(36\right)\left(7.2\right)\\ =\pi \left(12\right)\left(5.7\right)+\pi \left(36\right)\left(7.2\right)\\ =68.4\pi +259.2\pi \\ =327.6\pi \end{array}$

So, the volume of the small solid is $327.6\pi$ cubic centimeters.

Now, use the equal ratios to find the volume of the larger solid.

  $\Rightarrow \frac{V_{\text{small solid}}}{V_{\text{large solid}}}={\left(\frac{3}{4}\right)}^3$

  $\Rightarrow \frac{327.6\pi }{V_{\text{large solid}}}=\frac{27}{64}$

  $\Rightarrow V_{\text{large solid}}=\frac{327.6\pi \times 64}{27}$

  $\Rightarrow V_{\text{large solid}}=\frac{36.4\pi \times 64}{3}$

  $\Rightarrow V_{\text{large solid}}=\frac{36.4\pi \times 64}{3}$

  $\Rightarrow V_{\text{large solid}}\approx 2439.6$

Conclusion:

Therefore, the volume of the larger solid is about $2439.6$ cubic centimeters.