If the radius of a sphere is doubled, by what factor does its volume increase?

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Study for the Honors Mathematics 3 Exam. Engage with comprehensive flashcards and multiple choice questions, each with detailed explanations. Prepare and excel in your exam journey!

Multiple Choice

If the radius of a sphere is doubled, by what factor does its volume increase?

Explanation:
Volume scales with the cube of the radius. For a sphere, V = (4/3)πr^3. If the radius is doubled, the new radius is 2r, so the volume becomes V' = (4/3)π(2r)^3 = (4/3)π(8r^3) = 8V. This eightfold increase happens because you’re expanding in all three spatial directions, and the volume depends on the radius cubed, so doubling the radius multiplies volume by 2^3 = 8.

Volume scales with the cube of the radius. For a sphere, V = (4/3)πr^3. If the radius is doubled, the new radius is 2r, so the volume becomes V' = (4/3)π(2r)^3 = (4/3)π(8r^3) = 8V. This eightfold increase happens because you’re expanding in all three spatial directions, and the volume depends on the radius cubed, so doubling the radius multiplies volume by 2^3 = 8.

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