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Two solid right circular cones have the same height. The radii of their bases are $r 1$ and $r 2$ . They are melted and recast into a cylinder of same height. Then what is the radius of the base of the cylinder?

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2 r 12 + r 22 3

r 12 + r 22 2

r 12 + r 22 3

r 12 − r 22 2

answer is C.

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Detailed Solution

Given that two solid right circular cones have the same height, the radii of their bases are

r 1

and

r 2

and a cylinder of the same height.
We have the following formula,
Volume of the cone

= 13 π r 2 h

Volume of cylinder

= π r 2 h

We take

r

as the radius of the cylinder,

r 1

as the radius of the first cone and

r 2

as the radius of the second cone.
The volume of cone will be,
Volume of the two cones = volume of cylinder

13 π r 12 h + 13 π r 22 h = π r 2 h π h r 12 3 + r 22 3 = π r 2 h r 2 = r 2 + r 22 3 ⇒ r = r 12 + r 22 3

Thus, the radius is

r 12 + r 22 3

.
Therefore the correct option is 3.

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