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

Answer 1

The correct option is A

$25:64$

Step 1: Given data

$\frac{Volumeofcone1}{Volumeofcone2}=\frac{1}{4}$

$\frac{Diameterofcone1}{Diameterofcone2}=\frac{4}{5}$

$\frac{Heightofcone1}{Heightofcone2}=?$

Step 2: Calculating the ratio of the height of the cone $1$ to height of the cone $2$

We know that,

$$\begin{gathered} radius=\frac{diameter}{2} \\ \Rightarrow diameter=2\times radius \end{gathered}$$

Thus,

$\frac{d_1}{d_2}=\frac{2r_1}{2r_2}=\frac{r_1}{r_2}=\frac{4}{5}$

since the Volume of a cone$=\frac{1}{3}{\mathrm{\pi r}}^{2}h$

$$\begin{gathered} \frac{Volumeofcone1}{Volumeofcone2}=\frac{{\displaystyle \frac{1}{3}}\pi {\left(r_1\right)}^2{h}_1}{\frac{1}{3}\pi {\left(r_2\right)}^2{h}_2}={\left(\frac{r_1}{r_2}\right)}^2\times \frac{h_1}{h_2} \\ \Rightarrow \frac{1}{4}={\left(\frac{4}{5}\right)}^2\times \frac{h_1}{h_2} \\ \Rightarrow \frac{h_1}{h_2}=\frac{1}{4}\times \frac{25}{16}=\frac{25}{64} \end{gathered}$$

Hence, the ratio of the height of cone $1$ to the height of cone $2$ will be $25:64$

Hence, option A is correct.