Chapter 12.8, Problem 31HP

To determine

To find the height of cone C.

Answer to Problem 31HP

  $14$ cm

Explanation of Solution

Given:

  

Plane $P$ is parallel to the base of cone C.

Volume of the cone above the plane $=\frac{1}{8}$ volume of the cone C

Concept used:

If the two cones are similar, then the ratio of the volumes of the cones equals the cube of the ratio of their heights.

Calculation:

Since plane $P$ is parallel to the base of the cone C, the cone above the plane is similar to the cone C.

The ratio of the volumes of the cones will equal to the cube of the ratio of their heights.

Let $V_{\text{a}}$ and $h_{\text{a}}$ be respectively the volume and height of the cone above the plane.

  $\frac{V_{\text{a}}}{V_{\text{C}}}={\left(\frac{h_{\text{a}}}{h_{\text{C}}}\right)}^3$

  $\begin{array}{l}\Rightarrow \sqrt[3]{\frac{1}{8}}=\frac{7}{h_{\text{C}}}\\ \Rightarrow \frac{1}{2}=\frac{7}{h_{\text{C}}}\\ \Rightarrow h_{\text{C}}=14\end{array}$

Conclusion:

Therefore, the height of cone C is $14$ cm.