Question

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

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

Answer 1

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

Volume of a cone with radius 'r' and height 'h' is given by $\frac{1}{3}\pi r^{2}h.$
Given that, $\frac{r}{h}=\frac{1}{2}$
$\Rightarrow r=\frac{h}{2}$
So, volume of this cone = $\frac{1}{3}\pi (\frac{h}{2}{)}^{2}h$
= $\frac{1}{12}$ $\pi$$h^3$

Volume is given to be $144\pi$${cm}^3$.
$\Rightarrow \frac{1}{12}\pi h^3=144\pi$
$\Rightarrow h^3=144\times 12$
$\Rightarrow h=12cm\text{and}r=\frac{12}{2}=6cm$

Now, slant height = $\sqrt{h^2+r^2}$
= $\sqrt{{12}^2+6^2}$
= $\sqrt{180}$
= $6\sqrt{5}$ cm