If the volumes of two cones be in the ratio 1 : 4 and the radii of their bases be in the ratio 4 : 5

Question:

If the volumes of two cones be in the ratio 1 : 4 and the radii of their bases be in the ratio 4 : 5, then the ratio of their heights is
(a) 1 : 5
(b) 5 : 4
(c) 25 : 16
(d) 25 : 64

Solution:

(d) 25 : 64

Suppose that the radii of the cones are 4r and 5r and their heights are h and H, respectively .
It is given that the ratio of the volumes of the two cones is 1:4
Then we have:

$\frac{\frac{1}{3} \pi(4 r)^{2} h}{\frac{1}{3} \pi(5 r)^{2} H}=\frac{1}{4}$

$\Rightarrow \frac{16 r^{2} h}{25 r^{2} H}=\frac{1}{4}$

$\Rightarrow \frac{h}{H}=\frac{25}{64}$

$\therefore h: H=25: 64$

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