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Geometry and Trigonometry / Area and volume Difficulty: Hard

A hemisphere is half of a sphere. If a hemisphere has a radius of $27$ inches, which of the following is closest to the volume, in cubic inches, of this hemisphere?

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Explanation

Choice D is correct. The volume, $V$ , of a sphere is given by $V = \frac{4}{3} π r^{3}$ , where $r$ is the radius of the sphere. Since a hemisphere is half of a sphere, it follows that the volume, $V$ , of a hemisphere is given by $V = (\frac{1}{2}) (\frac{4}{3}) π r^{3}$ , or $V = \frac{2}{3} π r^{3}$ . Substituting $27$ for $r$ in this formula yields $upper V equals two thirds pi left parenthesis 27 right parenthesis cubed$ , which gives $upper V equals 13,122 pi$ , or $V$ is approximately equal to $41,223.98$ . Therefore, the choice that is closest to the volume, in cubic inches, of this hemisphere is $41,200$ .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.