Chapter 12.8, Problem 17PPS

To determine

To find: find the volume of pavilion.

Answer to Problem 17PPS

The volume of the pavilion is approximately $\text{150536}{\text{.32 ft}}^3$ .

Explanation of Solution

Given:

The diameter of a soccer ball is 8.7 in.

The volume of the ball is $344.8i{n}^3$ .

The diameter of the pavilion is 66 ft.

Calculation:

First convert feet into inches.

Use the fact that

  $1ft=12in.$

Thus,

  $\begin{array}{l}66ft=66x12in.\hfill \\ =792in.\hfill \end{array}$

Two solid figures are similar if they have the same shape and their corresponding linear measurements are proportional.

Since both the figures are sphere and their corresponding linear measurements are proportional, so the spheres are similar.

The ratio of the diameter of ball to pavilion is $\frac{8.7}{792}$ .

The scale factor is $\frac{8.7}{792}$ .

Now, use the property that if two solids are similar with scale factor of $\frac{a}{b}$ then the volume is in the ratio ${\left(\frac{a}{b}\right)}^3$ .

Thus,

  $\begin{array}{l}\frac{\text{Volume of soccer ball}}{\text{Volume of soccer ball shape pavilion}}={\left(\frac{a}{b}\right)}^3,\text{ where }\frac{a}{b}\text{ is the scale factor}\text{.}\\ \frac{344.8}{\text{Volume of soccer ball shape pavilion}}={\left(\frac{8.7}{792}\right)}^3\\ \frac{344.8}{\text{Volume of soccer ball shape pavilion}}=\frac{658.503}{496793088}\\ \left(658.503\right)\text{Volume of soccer ball shape pavilion}=344.8\times 496793088\\ \text{Volume of soccer ball shape pavilion}=\frac{344.8\times 496793088}{658.503}\\ \text{ = }\frac{171294256742.4}{658.503}\\ \approx 260126767.44\end{array}$

Therefore, the volume of the pavilion is $260126767.44{\text{ in}}^3$ .

Now, convert inches to feet.

Use the fact that

  $\begin{array}{l}1\text{ ft = 12 in}\text{.}\\ \text{1 in = }\frac{1}{12}\text{ft}\end{array}$

Thus,

  $\begin{array}{l}\text{260126767}{\text{.44 in}}^3=260126767.44\times \frac{1}{12}\times \frac{1}{12}\times \frac{1}{12}{\text{ft}}^3\\ =\frac{260126767.44}{1728}{\text{ ft}}^3\\ \approx \text{ 150536}{\text{.32 ft}}^3\end{array}$

Conclusion:

Therefore, the volume of the pavilion is approximately $\text{150536}{\text{.32 ft}}^3$ .