Question

A circle has a circumference of $6\pi \text{cm}$ and it is used as the base of a cylinder. If the cylinder has a total surface area of $30\pi {\text{cm}}^2$, then the volume of the cylinder is:

• A. $9\pi {\text{cm}}^3$
• B. $27\pi {\text{cm}}^3$
• C. $36\pi {\text{cm}}^3$
• D. $18\pi {\text{cm}}^3$

Answer 1

The correct option is D $18\pi {\text{cm}}^3$

Given, the circumference of the circle is $6\pi \text{cm}$ and total surface area of the cylinder is $30\pi {\text{cm}}^2$.

Let, the radius and height of the cylinder are $r$ and $h$ respectively.

Step 1: Circumference of the circle $=2\pi r$
$\Rightarrow 6\pi =2\pi r\left(\text{given}\right)$
Divide both sides by $2\pi$
$\Rightarrow \frac{6\pi }{2\pi }=\frac{2\pi r}{2\pi }$
$\Rightarrow 3=r\Rightarrow r=3\text{cm}$

Step 2: Total surface area of the cylinder
$=2\pi r^2+2\pi rh$
Putting $r=3$
$=2\pi (3{)}^2+2\pi \times 3\times h$
$=2\pi \times 9+6\pi h$
$=18\pi +6\pi h$
But total surface area is given as $30\pi$
$\Rightarrow 18\pi +6\pi h=30\pi$
Subtract $18\pi$ from both sides
$\Rightarrow 18\pi +6\pi h-18\pi =30\pi -18\pi$
$\Rightarrow 6\pi h=12\pi$
Divide both sides by $6\pi$
$\Rightarrow \frac{6\pi h}{6\pi }=\frac{12\pi }{6\pi }$
$\Rightarrow h=2\text{cm}$

Step 3: Volume of the cylinder is given by:
$V=\text{Area of the base}\times \text{height}$
$=\pi r^2\times h$
$=\pi (3{)}^2\times 2=\pi \times 9\times 2=18\pi {\text{cm}}^3$

Option (d.) is the correct choice.