Question

A cone of height 24 cm has a curved surface area 550 cm${}^2$. Find its volume.

Answer 1

Let r cm be the radius and l cm the slant height. Then,

$l^2=r^2+{24}^2$

$l^2=r^2+576$

$l=\sqrt{r^2+576}$

Curved surface area = 550 cm${}^2$

$\pi rl=550$

$\frac{22}{7}\times r\times \sqrt{r^2+576}=550$

$r\sqrt{r^2+576}=25\times 7$

$r^2(r^2+576)={25}^2\times 7^2$

$r^4+576r^2-(625\times 49)=0$

$(r^2+625)(r^2-49)=0$

$r=7$

Volume = $\frac{1}{3}\pi r^{2}h=\frac{1}{3}\times \frac{22}{7}\times 49\times 24=1232$ $c{m}^3$