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Geometry and Trigonometry / Area and volume Difficulty: Medium

A sphere has a radius of $StartFraction 17 Over 5 EndFraction$ feet. What is the volume, in cubic feet, of the sphere?

$StartFraction 5 pi Over 17 EndFraction$

$StartFraction 68 pi Over 15 EndFraction$

$StartFraction 32 pi Over 5 EndFraction$

$StartFraction 19,652 pi Over 375 EndFraction$

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Explanation

Choice D is correct. The volume, $V$ , of a sphere can be found using the formula $V = \frac{4}{3} π r^{3}$ , where $r$ is the radius of the sphere. It’s given that the sphere has a radius of $StartFraction 17 Over 5 EndFraction$ feet. Substituting $StartFraction 17 Over 5 EndFraction$ for $r$ in the formula $V = \frac{4}{3} π r^{3}$ yields $V = \frac{4}{3} π {(\frac{17}{5})}^{3}$ , which is equivalent to $V = \frac{4}{3} π (\frac{4,913}{125})$ , or $V = \frac{19,652 π}{375}$ . Therefore, the volume, in cubic feet, of the sphere is $StartFraction 19,652 pi Over 375 EndFraction$ .

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

Choice B is incorrect. This is the volume, in cubic feet, of a sphere with a radius of $RootIndex 3 StartRoot StartFraction 17 Over 5 EndFraction EndRoot$ feet.

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