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The radius of a sphere is three times the radius of the base of a cylinder. The height of the cylinder is nine times the radius of its base. If the total surface area of the cylinder and the numerical values of the volume of the sphere are equal, then what is the height of the cylinder?

The radius of a sphere is three times the radius of the base of a cylinder. The height of the cylinder is nine times the radius of its base. If the total surface area of the cylinder and the numerical values of the volume of the sphere are equal, then what is the height of the cylinder? Correct Answer 5 units

Given:

The radius of a sphere = three times the radius of the base of a cylinder.

The height of the cylinder = nine times the radius of its base.

The total surface area of the cylinder  = volume of the sphere 

Formula Used:

The total surface area of the cylinder = 2πr(r + h)

The volume of the sphere = (4/3)πr3

Calculation:

Let the radius of the base of a cylinder be r.

The radius of a sphere = 3r

The height of the cylinder = 9r

According to the question.

The total surface area of the cylinder  = Volume of the sphere

⇒ 2πr(r + h) = (4/3)πr3

⇒ 2 × r(r + 9r) = (4/3) (3r)3

⇒ 2 × 10r2 = (4/3) × 27r3

⇒ 20/(4 × 9) = r

⇒ r = 5/9

So, the height of the cylinder = 9r = 9 × (5/9) = 5 unit.

∴ The height of the cylinder is 5 unit.

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