Question : A sphere and another solid hemisphere have the same surface area. The ratio of their volumes is:
Option 1: $2 \sqrt{3}: 8$
Option 2: $3 \sqrt{3}: 8$
Option 3: $3 \sqrt{3}: 4$
Option 4: $\sqrt{3}: 4$
Correct Answer: $3 \sqrt{3}: 4$
Solution :
Let $R$ and $r$ be the radius of the solid sphere and the solid hemisphere respectively.
Now the surface area of the solid sphere and the solid hemisphere are $4\pi R^2$ and $3\pi r^2$
According to the problem,
$4\pi R^2 = 3\pi r^2$
$⇒4R^2 = 3r^2$
$⇒R =\frac{\sqrt{3}}{2}r$
Now ratio of their volumes is $\frac{\frac{4}{3}\pi R^3}{\frac{2}{3}\pi r^3} = \frac{2R^3}{r^3} =\frac{3\sqrt{3}r^3}{4r^3} =\frac{3\sqrt{3}}{4}$
Hence the correct answer is $3 \sqrt{3}: 4$.
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