Question

A hemisphere and a cone share the same base and have equal volumes. Given that their common radius is R, determine the height of the cone.

• A. 2R
• B. 4R
• C. 5R
• D. R

Answer 1

Answer:

A. 2R

A hemisphere and a cone have:

• Equal base radius = (R)

• Equal volumes

Use the volume formulas:

• Volume of hemisphere (= \frac{2}{3}\pi R^3)

• Volume of cone (= \frac{1}{3}\pi R^2 h)

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Since the volumes are equal,

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

Multiply both sides by 3:

$2\pi R^3=\pi R^2h$

Cancel (\pi R^2) from both sides:

$h=2R$

Answer:

$\boxed{h=2R}$

So, the height of the cone is twice its radius.