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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?
A. 2 units
B. 4 units
C. 6 units
D. 8 units

Respuesta :

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

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

Since, we know,

Volume of a cylinder = [tex]\pi r^{2}h[/tex]

i.e. [tex]24\pi=\pi r^{2}(3r)[/tex]

i.e. [tex]24=3r^{3}[/tex]

i.e. [tex]r^{3}=8[/tex]

i.e. r= ±2

Since, the radius cannot be negative.

So, the radius is 2 units.

Then, the height of the cylinder = [tex]3\times 2[/tex] = 6 units.

Hence, option C is correct.

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