Question

A cone of height of $8m$ has C.S.A $188.4m^2$. Find its volume $(\pi =3.14)$

Answer 1

Given height $h=8m$

C.S.A $=\pi rL=188.4m^2$

Slant height $L=\sqrt{h^2+r^2}$

$L=\sqrt{8^2+r^2}$

As $\pi rL=188.4$

$\pi \times r\times \sqrt{8^2+r^2}=188.4$

S.O.B.S

${\pi }^2\times r^2\times (8^2+r^2)=(188.4{)}^2$

$64{\pi }^2{r}^2+{\pi }^2{r}^4=35494.56$

Let $r^2=x$

${\pi }^2{x}^2+64{\pi }^{2}x-35494.56=0$

As $\pi =3.14,{\pi }^2=9.8596$

$9.8596(x^2+64x)=35494.56$

$x^2+64x=3600$

$x^2+64x-3600=0$

$x^2+100x-36x-3600=0$

$x(x+100)-36(x+100)=0$

$\therefore x=-100,36$

Take Positive Value

$x=36$ as $r^2=x$

$r^2=36$

$\therefore$ volume $=\frac{1}{3}\pi r^{2}h$

$=\frac{1}{3}\times 3.14\times 36\times 8$

$=\frac{904.32}{3}$

$=301.44m^3$