Question

A cone and a hemisphere have the same volume and base radius. Find the ratio of the height of the cone to its radius.

• A. 2 : 1
• B. 3 : 1
• C. 5 : 1
• D. 6 : 1

Answer 1

Answer:

A. 2 : 1

Let the common radius be (r), and cone height be (h).

Step 1: Write volumes

Hemisphere:
$V = \frac{2}{3}\pi r^3$

Cone:
$V = \frac{1}{3}\pi r^2 h$
Step 2: Equate volumes

$\frac{1}{3}\pi r^2 h = \frac{2}{3}\pi r^3$

Simplify

Cancel $(\frac{1}{3}\pi r^2):$

h = 2r

Required ratio

$h : r = 2 : 1$

Final Answer:

$\boxed{2 : 1}$