Discussion
The volume of a sphere is 27 times that of another sphere. What is the ratio of their surface areas?
Answer: A
The ratio of surface areas is 9 : 1. The volume of a sphere is proportional to the cube of its radius, so a volume ratio of 27 : 1 gives a radius ratio of 3 : 1. The surface area is proportional to the square of the radius, so the areas are in the ratio 9 : 1. 1 : 9 is inverted, 6 : 1 and 3 : 1 do not follow from squaring the radius ratio. The ratio of the radii is the cube root of 27 over the cube root of 1, which is 3 : 1. Surface area depends on the square of the radius, so the areas are in the ratio 9 : 1; the option 3 : 1 is the radii ratio, not the areas.
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