Question

The radius and height of a right circular cone are in th ratio $2:3$. Find the slant height if its volume is $100.48c{m}^3$. (Take $\pi =3.14$).

• A. $2\sqrt{13}cm$
• B. $4\sqrt{13}cm$
• C. $2\sqrt{3}cm$
• D. None of these

Answer 1

The correct option is A $2\sqrt{13}cm$

As the ratio of the radius and height of the cone is $2:3$

Let the radius of the cone be $2x$ and height be $3x$.

Given,

$V=100.48c{m}^3$.

We know that the volume of the cone

$V=\frac{1}{3}\pi r^{2}h$.

Then,

$\frac{1}{3}\pi (2x{)}^2\times 3x=100.48$

$⟹$ $3.14\times 4x^2\times x=100.48$

$⟹$$3.14\times 4x^3=100.48$

$⟹$$4x^3=32$

$⟹$$x^3=8$

$⟹$$x=2cm$.

Therefore,

$r=2\times 2=4cm$

$h=3\times 2=6cm$

We know that the slant height of the cone

$l=\sqrt{r^2+h^2}$ $=\sqrt{4^2+6^2}$

$=\sqrt{16+36}$ $=\sqrt{52}$

$=\sqrt{4\times 13}$ $⟹l=2\sqrt{13}cm$.

Hence, option $A$ is correct.