Chapter 12.8, Problem 23PPS

To determine

To determine the scale factor from the first sphere to the second.

Answer to Problem 23PPS

  $5.08$ to 1

Explanation of Solution

Given:

Two spheres are similar.

The radius of the first sphere is 10 feet.

The volume of the other sphere is 0.9 cubic meters.

2.54 cm = 1

Formula used:

Volume of sphere $=\frac{4}{3}\pi r^3$

Calculation:

At first changing the radius from feet to meter,

  $\Rightarrow \frac{10\text{ feet}}{1}\times \frac{12\text{ inch}}{1\text{ feet}}\times \frac{2.54\text{ centimeter}}{1\text{ inch}}\times \frac{1\text{ meter}}{100\text{ centimeter}}$

  $\Rightarrow 3.048\text{ meter}$

Now, volume of the first sphere $=\frac{4}{3}\pi r_1{}^3$

  $=\frac{4}{3}\times \pi \times {(3.048)}^3$

  $=\frac{4}{3}\times \pi \times {(3.048)}^3$

  $=\frac{4}{3}\times 88.959$

  $=118.613{\text{ meter}}^3$

  $\approx 118.6{\text{ meter}}^3$

The ratio of volume of sphere one and two gives the scale factor.

  $\Rightarrow \frac{\text{Volume of first sphere}}{\text{Volume of second sphere}}=\frac{118.6}{0.9}$

Multiplying 10 in both denominator and numerator,

  $\Rightarrow \frac{\text{Volume of first sphere}}{\text{Volume of second sphere}}=\frac{1186}{9}$

  $\Rightarrow \frac{\text{Volume of first sphere}}{\text{Volume of second sphere}}\approx \frac{131.7}{1}$

  $\Rightarrow \frac{\text{Volume of first sphere}}{\text{Volume of second sphere}}\approx \frac{{\left(5.08\right)}^3}{1}$

So, the scale factor is $5.08$ to 1.

Conclusion:

Therefore, the scale factor from the first sphere to the second is $5.08$ to 1.