If the radius of a sphere is halved, how does its volume change?

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Question: If the radius of a sphere is halved, how does its volume change?

Options:

Correct Answer: It reduces to one-eighth

Solution:

Volume of a sphere = (4/3)πr³. If radius is halved, volume becomes (4/3)π(1/2)³ = (4/3)π(1/8) = (1/6)π, which is one-eighth of the original volume.

If the radius of a sphere is halved, how does its volume change?

If the radius of a sphere is halved, how does its volume change?

Step 1: Write down the formula for the volume of a sphere, which is V = (4/3)πr³.
Step 2: Identify the original radius of the sphere as 'r'.
Step 3: If the radius is halved, the new radius becomes (1/2)r.
Step 4: Substitute the new radius into the volume formula: V = (4/3)π((1/2)r)³.
Step 5: Calculate the new volume: V = (4/3)π((1/2)³)(r³).
Step 6: Simplify (1/2)³ to get (1/8), so the volume becomes V = (4/3)π(1/8)(r³).
Step 7: Multiply (4/3) by (1/8) to get (4/24), which simplifies to (1/6).
Step 8: Therefore, the new volume is (1/6)πr³, which is one-eighth of the original volume.

Volume of a Sphere – Understanding the formula for the volume of a sphere and how changes in the radius affect the volume.
Exponential Relationships – Recognizing that volume changes with the cube of the radius, not linearly.