Question : The volume of the two cones is in the ratio 1 : 4 and their diameters are in the ratio 4 : 5. The ratio of their height is:
Option 1: 1 : 5
Option 2: 5 : 4
Option 3: 5 : 16
Option 4: 25 : 64
Correct Answer: 25 : 64
Solution :
The ratio of the volumes of two cones = 1 : 4
Ratio of their radii = Ratio of their diameters = 4 : 5
Let $h_1$ and $h_2$ be the heights and $r_1$ and $r_2$ be the radii.
Ratio of volumes = $\frac{\frac{1}{3} \pi (r_1)^2h_1}{\frac{1}{3} \pi (r_2)^2h_2}$
⇒ $\frac{1}{4}$ = $\frac{4^2\times h_1}{5^2 \times h_2}$
⇒ $\frac{h_1}{h_2}$ = $\frac{25}{64}$
Hence, the correct answer is 25 : 64.
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