Chapter 12.8, Problem 1BCYP

To determine

To compare the given cones and find their volume.

Answer to Problem 1BCYP

  $100\pi$ cubic inches, $172.8\pi$ cubic inches

Explanation of Solution

Given:

Two cones.

  

Concept used:

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

Formula used:

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

Calculation:

For the given two cones,

Match up the corresponding radii and heights.

  $\frac{\text{Small cone}}{\text{Large cone}}=\frac{5}{6}=\frac{12}{14.4}=0.833$

Here, the result is a number. It means that they are proportional.

So, congruent ratio shows that the given two cones are similar.

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

  $\begin{array}{l}=\frac{1}{3}\times \pi \times {\left(5\right)}^2\times 12\\ =\frac{1}{3}\times \pi \times 25\times 12\\ =100\pi \end{array}$

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

  $\begin{array}{l}=\frac{1}{3}\times \pi \times {\left(6\right)}^2\times 14.4\\ =\frac{1}{3}\times \pi \times 36\times 14.4\\ =172.8\pi \end{array}$

Conclusion:

Therefore, the given two cones are similar and their volume of first cone is $100\pi$ cubic inches and that of larger cone is $172.8\pi$ cubic inches.