Question

A solid is a cone standing on a hemisphere with both radii $2$ cm and the slant height of the cone $=2\sqrt{2}$ cm. Find the volume of the solid. (Use $\pi=3.14$)

Hint:

For a cone with given slant height $l$ and radius $r$, compute $h=\sqrt{l^2-r^2}$ first; then add volumes part-wise when solids are combined.

Answer 1

Step 1: Find the cone's height.
Given $r=2$ cm and $l=2\sqrt{2}$ cm, so \[ h=\sqrt{l^2-r^2}=\sqrt{(2\sqrt{2})^2-2^2}=\sqrt{8-4}=2\text{ cm}. \]

Step 2: Volumes.
Cone: $V_{\text{cone}}=\dfrac{1}{3}\pi r^2 h=\dfrac{1}{3}\pi\cdot 4\cdot 2=\dfrac{8}{3}\pi$.
Hemisphere: $V_{\text{hem}}=\dfrac{2}{3}\pi r^3=\dfrac{2}{3}\pi\cdot 8=\dfrac{16}{3}\pi$.
Total: \[ V=V_{\text{cone}}+V_{\text{hem}}=\frac{8}{3}\pi+\frac{16}{3}\pi=\frac{24}{3}\pi=8\pi. \] With $\pi=3.14$, \[ \boxed{V=8\pi=25.12\ \text{cm}^3}. \]