Question

If the volumes of two cones are in the ratio $1:4$ and their diameters are in the ratio $4:5$, then the ratio of their heights is ___________.

• A. $1:5$
• B. $5:4$
• C. $5:16$
• D. $25:64$

Answer 1

Answer: (D)

$25:64$

Let $V_1$ and $V_2$ are the volumes of the cones.

$\frac{V_1}{V_2}=\frac{1/3\pi r_1^2{h}_1}{1/3\pi r_2^2{h}_2}=\frac{r_1^2{h}_1}{r_2^2{h}_2}\Rightarrow \frac{1}{4}=\frac{4^2\times h_1}{5^2\times h_2}$ ($\because V_1:V_2=1:4$ and $r_1:r_2=4:5$)

Thus, $h_1:h_2=25:64$