Question

A cone of height $24$cm has a CSA of $550c{m}^2$. Find its volume.

Answer 1

$h=24cm,CSA=550c{m}^2$
We know that, $l^2=r^2+h^2=r^2+{24}^2=r^2+576$
$\therefore l=\sqrt{r^2+576}$
CSA of the cone$=550c{m}^2$
$\therefore \pi rl=550$
$\frac{22}{7}\times r\times \sqrt{r^2+576}=550$
$\therefore r\sqrt{r^2+576}=\frac{550\times 7}{22}$
$\therefore r\sqrt{r^2+576}=175$
Square on both sides
${\left(r\sqrt{r^2+576}\right)}^2={\left(175\right)}^2$
$\therefore r^2(r^2+576)=30625$
$r^4+576r^2-30625=0$

Factorising this equation we get,
$(r^2+625)(r^2-49)=0$

If $r^2+625=0,r^2=-625$, It is not possible

If $r^2-49=0,r^2=49,r=\sqrt{49}\therefore r=7cm$
Volume of the cone $=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\times \frac{22}{7}\times 7\times 7\times 24c{m}^3$
$\therefore$ Volume of the cone$=1232c{m}^3$.