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

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

Options:

Correct Answer: It increases by a factor of eight

Solution:

Volume of a sphere = (4/3)πr³. If r is doubled, the new volume = (4/3)π(2r)³ = (4/3)π(8r³) = 8 times the original volume.

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

If the radius of a sphere is doubled, how does the 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 doubled, the new radius becomes '2r'.
Step 4: Substitute the new radius into the volume formula: V = (4/3)π(2r)³.
Step 5: Calculate (2r)³, which is 2³ * r³ = 8r³.
Step 6: Replace (2r)³ in the volume formula: V = (4/3)π(8r³).
Step 7: Simplify the equation: V = (4/3) * 8 * π * r³ = (32/3)πr³.
Step 8: Compare the new volume (32/3)πr³ with the original volume (4/3)πr³.
Step 9: Notice that (32/3) is 8 times (4/3), so the new volume is 8 times 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 Growth – Recognizing that volume changes with the cube of the radius, illustrating the concept of exponential growth.