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

Answer 1

(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:
$$\begin{gathered} \frac{{\displaystyle \frac{1}{3}}\mathrm{\pi }{\left(4r\right)}^{2}h}{{\displaystyle \frac{1}{3}}\mathrm{\pi }{\left(5r\right)}^{2}H}=\frac{1}{4} \\ \Rightarrow \frac{16r^{\mathit{2}}h}{25r^{2}H}=\frac{1}{4} \\ \Rightarrow \frac{h}{H}=\frac{25}{64} \\ \therefore \mathit{}h:H=25:64 \end{gathered}$$