Q 6. - Calculus

Question

Give formulas for the volume and the surface area of a cone whose height $h$ is three times its radius $r$. 

Step-by-Step Solution

Answer

Volume of cone is $V=\pi r^3$.

Surface area of cone is $S=\pi r^2\left(\sqrt{10}+1\right)$.

Step 1. Standard formulas

The volume and surface area of a right circular cone of radius $r$ and height $h$ are

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

$S=\pi r\sqrt{r^2+h^2}+\pi r^2$

Step 2. Substitution and solution

Now given that the radius is $r$ and the height is three times its radius means $h=3r$ of cone

Substitute in standard formulas we get, 

Volume of cone is $V=\frac{1}{3}\pi r^2\left(3r\right)=\pi r^3$.

Surface area of cone is $S=\pi r\sqrt{r^2+{\left(3r\right)}^2}+\pi r^2=\sqrt{10}\pi r^2+\pi r^2=\pi r^2\left(\sqrt{10}+1\right)$.