Question
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A right circular cylinder has a volume of $81\pi .$If the circumference of the base is $6\pi ,$what is the height?
A. 6
B. 9
C. 12
D. 15

Answer Verified Verified
Hint: Here, first calculate the radius of the base as it circumference is given. Then apply the formula of volume of the right circular cylinder by assuming height h and equate that volume to the given volume of the cylinder. By solving the obtained equation we can get the value of the height of the cylinder.

Complete step by step answer:
Given, a right circular cylinder has a volume $81\pi .$
And the circumference of the base is $6\pi .$
Let r be the radius of the circular base and h be the height of the cylinder.
The base of the right circular cylinder is a circle in shape.
Given, the circumference of the circular base is $6\pi .$
[Circumference of circle = $2\pi r,$ where r is the radius of the circle]
$
  \therefore 2\pi r = 6\pi \\
   \Rightarrow r = \dfrac{{6\pi }}{{2\pi }} = 3 \\
 $
Therefore, radius of right circular cylinder = 3 units
The volume of the cylinder with radius r and height h is given as $\pi {r^2}h$ .
$
  \therefore \pi {r^2}h = 81\pi \\
   \Rightarrow {(3)^2}h = \dfrac{{81\pi }}{\pi } = 81 \\
   \Rightarrow h = \dfrac{{81}}{9} = 9 \\
 $
Therefore, the height of the cylinder is 9 units.

Hence, (B) is correct.

Note:
 In these types of questions, find the value of unknown things with the help of given value, by assuming radius and height as variable quantity. You must the formula of volumes and circumferences of basic shapes to solve these types of questions as these are directly based on a formula. Never put a formula containing more than one variable or you must have two equations for finding two variables. In this case, if you first apply the volume formula of the cylinder without finding the radius, it will be useless.
Alternatively, first, find the radius of the base, and then find the area of the base. Apply formula, Volume of cylinder = Area of circular base × Height, as the volume is given we can easily find the height of the cylinder by dividing the given area by area of the base.

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