Question

The height of a cone is 50 cm. A small cone is cut off at the top by a plane parallel to its base. If the volume of the smaller conical portion is $\frac{8}{125}$ times the volume of the given cone, then the ratio of the height of the frustum so formed to the height of original cone is

• A. $\frac{4}{5}$
• B. $\frac{2}{5}$
• C. $\frac{7}{9}$
• D. $\frac{3}{5}$

Answer 1

The correct option is D $\frac{3}{5}$


Let r and R be the radius of lower and upper ends of the frustum respectively and h be the height of the frustum.

Volume of smaller cone $=\frac{8}{125}$ × Volume of given cone

$\Rightarrow \frac{1}{3}\pi r^2(50-h)=\frac{8}{125}\times \frac{1}{3}\times \pi \times R^2\left(50\right)$

$\Rightarrow \frac{r^2}{R^2}=\frac{80}{25(50-h)}$

$\Rightarrow \frac{r^2}{R^2}=\frac{16}{5(50-h)}$ ...(i)

Also, the smaller and the given cones are similar,

$\Rightarrow \frac{r}{R}=\frac{50-h}{50}$ ...(ii)

From (i) and (ii), we get

$\frac{(50-h{)}^2}{2500}=\frac{16}{5(50-h)}$

$\Rightarrow (50-h{)}^3=16\times 500$

$\Rightarrow (50-h{)}^3=8000$

$\Rightarrow (50-h)=20$

$\Rightarrow h=30cm$

$\therefore$ Ratio of the height of frustum to the height of original cone $=\frac{30}{50}$

$=\frac{3}{5}$

= 3 : 5

Hence, the correct answer is option (d).