Question

The radius and height of a right circular cone are in the ratio $4:3$ and its volume is $2156c{m}^3$. Find the curved surface area of the cone.

Answer 1

Let the radius and height of the cone be $4xcm$ and $3xcm,$ respectively.
Volume of the right circular cone $=2156c{m}^3$
We have:
$\frac{1}{3}\pi (4x{)}^2\times (3x)=2156$ [$\because$ Volume of cone $=\frac{1}{3}\pi r^{2}h$]

$\Rightarrow 16x^3\pi =2156$

$\Rightarrow x^3=\frac{2156\times 7}{16\times 22}$

$\Rightarrow x^3=\frac{196\times 7}{16\times 2}=\frac{49\times 7}{4\times 2}$

$\Rightarrow x^3=\frac{7\times 7\times 7}{2\times 2\times 2}$

$\Rightarrow x=\frac{7}{2}=3.5$
Now, radius of the cone, $r=4x=4\times 3.5=14cm;$ height, $h=3x=3\times 3.5=10.5cm$
$\therefore$ Curved surface area of the cone $=\pi rl$

$=\frac{22}{7}\times 14\times \sqrt{(14{)}^2+(10.5{)}^2}$ [$\because$ Slant height, $l=\sqrt{r^2+h^2}$]

$=22\times 2\times \sqrt{196+110.25}$

$=44\times 17.5$

$=770c{m}^2$