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Geometry and Trigonometry / Area and volume Difficulty: Hard

A cube has an edge length of $68$ inches. A solid sphere with a radius of $34$ inches is inside the cube, such that the sphere touches the center of each face of the cube. To the nearest cubic inch, what is the volume of the space in the cube not taken up by the sphere?

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Explanation

Choice A is correct. The volume of a cube can be found by using the formula $V = s^{3}$ , where $V$ is the volume and $s$ is the edge length of the cube. Therefore, the volume of the given cube is $upper V equals 68 cubed$ , or $314,432$ cubic inches. The volume of a sphere can be found by using the formula $V = \frac{4}{3} π r^{3}$ , where $V$ is the volume and $r$ is the radius of the sphere. Therefore, the volume of the given sphere is $upper V equals four thirds pi left parenthesis 34 right parenthesis cubed$ , or approximately $164,636$ cubic inches. The volume of the space in the cube not taken up by the sphere is the difference between the volume of the cube and volume of the sphere. Subtracting the approximate volume of the sphere from the volume of the cube gives $314,432 - 164,636 = 149,796$ cubic inches.

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

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

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