Question : If the ratio of the diameters of two right circular cones of equal height is 3 : 4, then the ratio of their volume will be:
Option 1: 3 : 4
Option 2: 9 : 16
Option 3: 16 : 9
Option 4: 27 : 64
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Correct Answer: 9 : 16
Solution :
The ratio of the diameters of two right circular cones of equal height = 3 : 4
Let diameter of 1
st
cone be $3x$.
Diameter of 2
nd
cone $= 4x$
Radius of the 1
st
cone, $r_1 = \frac{3x}{2}$
Radius of the 2
nd
cone, $r_2 = \frac{4x}{2}$ = $2x$
and $h_1 = h_2$
Ratio of the volumes of the cones = $\frac{\frac{1}{3}\pi r_{1}^{2}h_1}{\frac{1}{3}\pi r_{2}^{2}h_2} = \frac{r_{1}^{2}}{ r_{2}^{2}} = \frac{\frac{9}{4}}{4} = \frac{9}{16}$
Hence, the correct answer is 9 : 16.
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