Question

The radius and height of a cone are in the ratio 1:2. If its volume is 144π cm${}^3$, its slant height = .

• A. 5$\sqrt{5}$ cm
• B. 6$\sqrt{5}$ cm
• C. 4$\sqrt{5}$ cm

Answer 1

The correct option is B 6$\sqrt{5}$ cm

Volume of a cone = $\frac{1}{3}$ $\pi$$r^{2}h$
Given that $\frac{r}{h}=\frac{1}{2}$
$\Rightarrow r=\frac{h}{2}$.
Therefore, volume
= $\frac{1}{3}\pi (\frac{h}{2}{)}^2\times h$
= $\frac{1}{12}\pi h^3$
Volume is given to be $144\pi$ ${cm}^3$.
$\Rightarrow \frac{1}{12}\pi h^3$ = 144 $\pi$
$\Rightarrow$ $h^3$ = 144 $\times$ 12
$\Rightarrow h=12cm$
$\Rightarrow r=6cm$
Now, slant height
=$\sqrt{h^2+r^2}$
= $\sqrt{{12}^2+6^2}$
=$\sqrt{180}$
= 6 $\sqrt{5}$ cm
Therefore, slant height is $6\sqrt{5}$ cm long.