Question

A circle has a circumference of 6π centimeters and is used as the base of a cylinder. If the cylinder has a total surface area of 30π square centimeters, what will the volume of the cylinder be?

• A. $9\pi$ cu cm
• B. $27\pi$ cu cm
• C. $36\pi$ cu cm
• D. $18\pi$ cu cm

Answer 1

The correct option is D $18\pi$ cu cm

Given, the circumference of the circle is $6\pi$ cm and total surface area of the cylinder is $30\pi$ sq cm.

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

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

$\text{Step 2:}$ Total surface area of the cylinder
$=2\pi r^2+2\pi rh$
Putting $r=3,$
$=2\pi (3{)}^2+2\pi \times 3\times h$
$=2\pi \times 9+6\pi h$
$=18\pi +6\pi h$ sq cm

But the total surface area is given as $30\pi$.
$18\pi +6\pi h=30\pi$
Subtract $18\pi$ from both sides.
$18\pi +6\pi h-18\pi =30\pi -18\pi$
$6\pi h=12\pi$
Divide both sides by $6\pi .$
$\Rightarrow \frac{6\pi h}{6\pi }=\frac{12\pi }{6\pi }$
$\Rightarrow h=2$ cm

$\text{Step 3:}$ Volume of the cylinder
V = area of the base × height
$=\pi r^{2}h$
$=\pi \left(3{)}^2\right(2)$
$=\pi \times 9\times 2$
$=18\pi$ cu cm