The ratio of diameters for two spheres is 1:2. What is the ratio of their volumes?
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Answer: C. 1:8
Explanation
The ratio of the diameters of two spheres is directly proportional to the ratio of their radii, which is one to two. The volume of a sphere is proportional to the cube of its radius according to the standard geometric formula four-thirds pi r cubed. Cubing the radii ratio of one to two gives one cubed to two cubed, resulting in a volume ratio of one to eight. Geometric scaling laws dictate that when linear dimensions scale by a factor k, surface areas scale by k squared, and volumes scale by k cubed. The radius of a sphere is exactly half of its diameter, meaning diameter ratios equal radius ratios. Volume calculations for three-dimensional shapes involve three spatial dimensions multiplied together.
Option A represents an incorrect linear or surface area ratio rather than the three-dimensional volume ratio. Option B reflects the squared area scaling factor of four-thirds pi r squared rather than the cubed volume factor. Option D incorrectly suggests a linear ratio that ignores the cubic power law governing three-dimensional volumes.
- Subject
- Mathematics
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- Mathematics
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- —
- Difficulty
- medium
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