Q 5. - Calculus

Question

Give formulas for the volume and the surface area of a cylinder whose radius $r$ is half of its height $h$. 

Step-by-Step Solution

Answer

Volume of cylinder is $V=\frac{\pi h^3}{4}$.

Surface area of cylinder is $S=\frac{3\pi h^2}{2}$.

Step 1. Standard formulas

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

$V=\pi r^{2}h$

$S=2\pi rh+2\pi r^2$

Step 2. Substitution and solution

Now given that the height is $h$ and radius is half of its height means $r=\frac{h}{2}$ of cylinder

Substitute in standard formulas we get,

Volume of cylinder is $V=\pi {\left(\frac{h}{2}\right)}^{2}h=\frac{\pi h^3}{4}$.

Surface area of cylinder is $S=2\pi \frac{h}{2}h+2\pi {\left(\frac{h}{2}\right)}^2=\pi h^2+\frac{\pi h^2}{2}=\frac{3\pi h^2}{2}$