Question

The radius and the height of a right circular cone are in the ratio of $3:5$. If its volume is $120$ $\pi$ cubic m, then its slant height is:

• A. $3\sqrt{34}\text{m}$
• B. $2\sqrt{28}\text{m}$
• C. $2\sqrt{44}\text{m}$
• D. $2\sqrt{34}\text{m}$

Answer 1

Answer: (D)

$2\sqrt{34}\text{m}$

Given, the ratio of the base radius and the height of the right circular cone $=3:5$.

Suppose the base radius and the height of the right circular cone are $3x$ m and $5x$ m, respectively.

$\therefore$ Volume of the cone $=\frac{1}{3}\pi r^{2}h$ $=\frac{1}{3}\pi {\left(3x\right)}^2\times 5x$.

According to question:

Volume of the cone$=120\pi {\text{m}}^3$

$\therefore \frac{1}{3}\pi \times 9x^2\times 5x=120\pi$

$x^3=\frac{120\times 3}{9\times 5}$

$x^3=8$

$x^3={\left(2\right)}^3\Rightarrow x=2\text{m}$.

$\therefore$ The radius and the height of the cone will be $3\times 2=6\text{m}$ and $5\times 2=10\text{m}$, respectively.

$\therefore$ Slant height of the cone $l=\sqrt{r^2+h^2}$

$=\sqrt{{\left(6\right)}^2+{\left(10\right)}^2}$

$=\sqrt{36+100}$

$=\sqrt{136}=2\sqrt{34}\text{m}$.

Hence, option $D$ is correct.