The ratio of surface areas for two spheres is 1:4. What is the ratio of their volumes?
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Answer: B. 1:8
Explanation
The ratio of the surface areas of two spheres equals the square of the ratio of their radii. Taking the square root of the surface area ratio gives a radius ratio of one to two. Cubing the radius ratio determines the final volume ratio of one to eight for the spheres. Geometric scaling laws state that linear dimensions scale by factor k, areas by k squared, and volumes by k cubed. Similar three dimensional figures maintain constant proportions across all corresponding linear features. Sphere surface area is proportional to the square of radius while volume is proportional to the cube.
Scaling principles are vital in physics and engineering when analyzing models versus full scale prototypes. Astrophysicists use spherical geometry ratios to compare planetary masses, volumes, and surface characteristics. Manufacturing industries apply scaling laws to determine material requirements for containers of different sizes. Option A incorrectly assumes the volume ratio remains identical to the surface area ratio. Option B mistakenly squares the surface area ratio instead of properly cubing the radius ratio. Option D applies an incorrect exponential scaling factor that overestimates the volume difference.
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- Mathematics
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- Mathematics
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