Question

The total surface area of a right circular cone of slant height 13 cm is 90$\pi$ $c{m}^2$. Calculate :
(i) its radius in cm,
(ii) its volume in $c{m}^3$, in terms of $\pi$.

Answer 1

Given : slant height , $l$=13cm.

Let, radius = r cm and height = h cm.

(i) Total surface area = $\pi r(l+r)=\left[\pi r\right(13+r\left)\right]c{m}^2$.

$\pi r(13+r)=90\pi$

$r^2+13r-90=0$

$r^2+18r-5r-90=0$

$r(r+18)-5(r+18)=0$

$(r+18)(r-5)=0$

$r=5,r=-18$

$r=5$ [Neglecting r=-18, as radius cannot be negative]

Radius of the cone $=5cm$.

(ii) Since, h = $\sqrt{l^2-r^2}-\sqrt{(13{)}^2-(5{)}^2}$

= $\sqrt{{169}^2-{25}^2}=\sqrt{144}$ = 12 cm.

$\therefore$ Volume of the cone = $\frac{1}{3}\pi r^{2}h=(\frac{1}{3}\times \pi \times 5\times 5\times 12)c{m}^3$

$=25\pi \times 4$

= 100 $\pi c{m}^3$.