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Difficulty: HardVolume and Surface Area of Solids

A solid metal right circular cylinder has a base radius of 12 centimeters12\text{ centimeters} and a height of 3 centimeters3\text{ centimeters}. A cone with the same base and height as the cylinder is carved out and removed. The remaining metal is melted and recast into a solid sphere. What is the radius, in centimeters, of the sphere?

Answer: 6 centimeters

Answer

The radius of the sphere is 6 centimeters.
The volume of a cylinder is V=πr2h=π(12)2(3)=432πV = \pi r^2 h = \pi (12)^2 (3) = 432\pi. The volume of the carved-out cone is V=13πr2h=13π(12)2(3)=144πV = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi (12)^2 (3) = 144\pi. Subtracting the cone's volume from the cylinder's volume yields the remaining metal volume: 432π−144π=288π432\pi - 144\pi = 288\pi. The volume of the recast sphere is 43πR3=288π\frac{4}{3}\pi R^3 = 288\pi. Dividing by π\pi and multiplying by 34\frac{3}{4} gives R3=216R^3 = 216. Taking the cube root of both sides gives the radius R=6R = 6.

Step-by-Step Solution

1
Calculate the volume of the original cylinder.
432π cubic centimeters432\pi\text{ cubic centimeters}
This determines the starting volume of the solid metal block before any material is removed.
2
Calculate the volume of the cone that is carved out.
144π cubic centimeters144\pi\text{ cubic centimeters}
This determines how much metal is discarded from the cylinder.
3
Subtract the cone's volume from the cylinder's volume.
288π cubic centimeters288\pi\text{ cubic centimeters}
This gives the volume of the remaining metal that will be melted and recast.
4
Equate the remaining volume to the volume formula of a sphere and solve for the radius.
6 centimeters6\text{ centimeters}
This yields the radius of the newly formed sphere.

Key Concept

Volume of composite solids and conservation of volume during recasting
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Topics

Volume and Surface Area of Solids

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