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A solid is in the form of a right circular cone mounted on a solid hemisphere with the same radius is made from a piece of metal. The radius of the hemisphere is $\frac{1}{3}$ of the vertical height of the conical part. If the radius of the base of the cone is $r$ , the volume of the piece of metal is

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\frac{11}{3} π r^{3}

\frac{8}{3} π r^{3}

\frac{7}{3} π r^{3}

\frac{5}{3} π r^{3}

answer is D.

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Detailed Solution

Given radius of the hemisphere is

\frac{1}{3}

of the vertical height of the conical part and the radius of the base of the cone is

r

.
Volume of a hemisphere of radius,

^{'} r^{'}

= \frac{2}{3} π r^{3}

Volume of a cone

= \frac{1}{3} π r^{2} h

, where

r

is the radius of the base of the cone and

h

is the height.
As,

r

= \frac{1}{3} h

\Rightarrow h = 3 r

Volume of the solid

=

Volume of the hemispherical part

+

Volume of conical part

\Rightarrow (\frac{2}{3} π r^{3}) + (\frac{1}{3} \times π \times r^{2} \times 3 r)

\Rightarrow \frac{2}{3} π r^{3} + π r^{3}

\Rightarrow \frac{5}{3} π r^{3}

Therefore, volume of the solid is

\frac{5}{3} π r^{3}

.
Hence, the correct option is 4.

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