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A hollow cylinder is formed by cutting out a solid cylinder of radius r/k from a solid cylinder of radius r. The height of both solid cylinders was equal. The volume of the hollow cylinder m times the volume of the new solid cylinder. What is the relationship between m and k?

A hollow cylinder is formed by cutting out a solid cylinder of radius r/k from a solid cylinder of radius r. The height of both solid cylinders was equal. The volume of the hollow cylinder m times the volume of the new solid cylinder. What is the relationship between m and k? Correct Answer k<sup>2</sup> = m + 1

Calculation:

Volume of old cylinder = πr2h

Volume of new solid cylinder = πr2h/k2

Volume of hollow cylinder = πr2h - (πr2h/k2)

⇒ πr2h(1 - 1/k2)

∴ πr2h(1 - 1/k2) = m × πr2h/k2

⇒ 1 - 1/k2 = m/k2

⇒ (m + 1)/k2 = 1

⇒ k2 = m + 1

Additional Information

The total surface area of a cylinder is the sum of curved surface area and base areas.

The volume of the cylinder is the product of the base area and height of the cylinder.

The total surface area of a circular cylinder with height h and base radius r given by 2πr(r + h).

The curved surface area of a circular cylinder is equal to 2πrh.

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