A cone and a hemisphere have the same radius. Their combined height is 24 cm. If both have equal volumes, find the radius.

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Accepted Answer

1. Let the common radius be $r$. The hemisphere height is $r$, so cone height is $24-r$.

2. Cone volume: $V_{cone}=\frac{1}{3}\pi r^2(24-r)$. Hemisphere volume: $V_{hem}=\frac{2}{3}\pi r^3$.

3. Set equal: $\frac{1}{3}\pi r^2(24-r)=\frac{2}{3}\pi r^3$. Cancel common terms to get $24-r=2r$.

4. Solve: $24=3r$, so $r=8$. Thus the radius is 8 cm, option **(a)**.

FINAL_ANSWER: <a>

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