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A solid cylinder of diameter $12 cm$ and height $15 cm$ is melted and recast into 12 toys in the shape of a right circular cone mounted on a hemisphere. Find the radius of the hemisphere and total height of the toy if height of the cone is 3 times the radius.

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2cm and 8cm

3cm and 9cm

4cm and 10cm

5cm and 11cm

answer is B.

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Detailed Solution

Given diameter of solid cylinder is 12cm, height is 15cm.
For toy, height of cone is 3 times the radius.
The volume of the cylinder is

V = π r 2 h

, here r is radius and h is height.
The volume of the hemisphere is

V = 23 π r 3

, here r is radius.
Let the radius of the hemisphere is

R

.
So,

Volume of the cylinder = 12 × Volume of a toy ⇒ π r 2 h = 12 13 π R 2 H + 23 π R 3 ⇒ π r 2 h = 123 π R 2 3 R + 2 R 3 ⇒ 62 × 15 = 4 3 R 3 + 2 R 3

Therefore,

⇒ 20 R 3 = 540 ⇒ R 3 = 27 ⇒ R = 3 cm

The total height of the toy,

H = 3 × R ⇒ H = 3 × 3 ⇒ H = 9 cm

Thus, the radius of the hemisphere is 3 cm and the total height of the toy is 9 cm.
Therefore, option (2) is correct.

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