If the radius of a sphere is increased by 100% (i.e., doubled), by what percentage does its volume increase?

Introduction / Context:Volumes scale with the cube of a linear dimension. Doubling a radius multiplies the volume by 2^3 = 8 times the original volume.

Given Data / Assumptions:

• Initial radius r, new radius r′ = 2r.
• Sphere volume V = (4/3)πr^3.

Concept / Approach:Compute V′/V = (r′/r)^3 = 2^3 = 8, then convert to a percentage increase over the original (i.e., compared to 1×).

Step-by-Step Solution:

Verification / Alternative check:Plug r = 1 into V = (4/3)π; doubling to r = 2 gives V′ = (4/3)π * 8 = 8V → +700%.

Why Other Options Are Wrong:

• 300%, 500%, 900%: Do not match the cubic scaling for doubling.

Common Pitfalls:Confusing area scaling (square) with volume scaling (cube).

Final Answer:700%