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The height of a right cylinder is 3 times the radius of the base. The volume of the cylinder is 24π cubic units. What is the height of the cylinder?

A2 units
B4 units
C6 units
D8 units

Verified Answer :

carlosego

By definition, the volume of a cylinder is given by:
 \(V = \pi * r ^ 2 * h\)
 Where,
 r: cylinder radius
 h: cylinder height
 On the other hand we have:
 The height is 3 times the radius:
 \(h = 3r\)
 Therefore, rewriting the volume as a function of the height we have:
 \(V = \pi * (\frac{h}{3}) ^ 2 * h\)
 \(V = ( \frac{1}{9} ) \pi * h ^ 3\)
 From here, we clear h.
 We have then:
 \(h ^ 3 = \frac{9V}{ \pi}\)
 \(h = \sqrt[3]{\frac{9V}{ \pi}}\)
 Substituting values we have:
 \(h = \sqrt[3]{\frac{9(24\pi)}{ \pi}}\)
 \(h = \sqrt[3]{9*24}\)
 \(h = \sqrt[3]{216}\)
 \(h = 6\)
 Answer:
 the height of the cylinder is:
 C. 6 units

Answer:

C. 6 units

Step-by-step explanation:

Let, the radius of the cylinder = r units.

Then, the height of the cylinder = \(3\times r\) = \(3r\) units.

Also, the volume of the cylinder is 24π cubic units.

Since, we know,

Volume of a cylinder = \(\pi r^{2}h\)

i.e. \(24\pi=\pi r^{2}(3r)\)

i.e. \(24=3r^{3}\)

i.e. \(r^{3}=8\)

i.e. r= ±2

Since, the radius cannot be negative.

So, the radius is 2 units.

Then, the height of the cylinder = \(3\times 2\) = 6 units.

Hence, option C is correct.

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