Chapter 12.8, Problem 22PPS

To determine

To determine the volume of the other cylinder.

Answer to Problem 22PPS

  $\approx \text{380}\text{.6}\pi {\text{ centimeter}}^3$

Explanation of Solution

Given:

Two cylinders are similar.

The height of the first cylinder is 23 centimeter and the height of the other cylinder is 8 inch. The volume of the first cylinder is 552p cubic centimeter.

2.54 centimeter = 1 inch

Formula used:

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

  $\left[r=\text{radius and }h=\text{height}\right]$

Calculation:

At first determining the scale factor,

  $\Rightarrow \frac{\text{height of first cylinder}}{\text{height of second cylinder}}=\frac{23}{8\times \left(2.54\right)}$

  $\Rightarrow \frac{\text{height of first cylinder}}{\text{height of second cylinder}}=\frac{23}{20.32}$

  $\Rightarrow \frac{\text{height of first cylinder}}{\text{height of second cylinder}}=\frac{2300}{2032}$

  $\Rightarrow \frac{\text{height of first cylinder}}{\text{height of second cylinder}}=\frac{575}{508}$

Now, find the radius of the first cylinder.

Volume of cylinder $=\pi r_1{}^2{h}_1$

  $=\pi \times {(r)}^2\times 23$

  $\Rightarrow$

  $552\pi =\pi \times {(r_1)}^2\times 23$

  $\Rightarrow$

  $r=\sqrt{\frac{552}{23}}$

  $\Rightarrow$

  $r_1=2\sqrt{6}$

Now, the radius of the second can be find by using scale factor,

  $\Rightarrow \frac{\text{radius of first cylinder}}{\text{radius of second cylinder}}=\frac{575}{508}$

  $\Rightarrow \frac{2\sqrt{6}}{\text{radius of second cylinder}}=\frac{575}{508}$

  $\Rightarrow \text{radius of second cylinder}=\frac{508\times 2\sqrt{6}}{575}$

  $\Rightarrow \text{radius of second cylinder}\approx \text{4}\text{.328}$

Now, find the volume of second cylinder.

Volume of second cylinder $=\pi r_2{}^2{h}_2$

  $=\pi \times {(4.328)}^2\times 20.32$

  $=380.625\pi$

  $\approx \text{380}\text{.6}\pi {\text{ centimeter}}^3$

Conclusion:

Therefore, the volume of the other cylinder $\approx \text{380}\text{.6}\pi {\text{ centimeter}}^3$ .