Question : If the radius of a sphere is $\frac{3}{4}$th of the radius of a hemisphere, then what will be the ratio of the volumes of sphere and hemisphere?
Option 1: 9 : 16
Option 2: 51 : 64
Option 3: 27 : 32
Option 4: 18 : 64
Correct Answer: 27 : 32
Solution :
Let the radius of the hemisphere be $r$.
Given: Radius of sphere = $\frac{3}{4}$th of the radius of hemisphere
Volume of a sphere = $\frac{4}{3} \pi (\frac{3}{4} r)^3 = \frac{9\pi }{16} r^3$
Volume of a hemisphere = $\frac{2}{3} \pi r^3$
The ratio of the volumes of the sphere to the hemisphere is:
$\frac{{\text{volume of a sphere}}}{{\text{volume of a sphere}}} = \frac{\frac{9\pi }{16} r^3}{\frac{2}{3} \pi r^3} = \frac{27}{32}$
Hence, the correct answer is 27 : 32.
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