Question:
If the percentage change in the radius of a sphere is \(2\%\), then find the percentage change in the volume of the sphere.
If the percentage change in the radius of a sphere is \(2\%\), then find the percentage change in the volume of the sphere.
Updated On: Apr 6, 2026
- \(2\%\)
- \(4\%\)
- \(6\%\)
- \(8\%\)
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The Correct Option is C
Solution and Explanation
Concept:
The volume of a sphere is given by
\[
V = \frac{4}{3}\pi r^3
\]
Thus, volume is proportional to the cube of the radius:
\[
V \propto r^3
\]
For small percentage changes, the percentage change in a quantity raised to a power follows
\[
\frac{\Delta V}{V} = 3 \frac{\Delta r}{r}
\]
Step 1: {\color{red}Use the relation between percentage changes.}
\[
\%\Delta V = 3 \times \%\Delta r
\]
Step 2: {\color{red}Substitute the given percentage change in radius.}
\[
\%\Delta V = 3 \times 2\%
\]
\[
\%\Delta V = 6\%
\]
Hence, the percentage change in the volume of the sphere is
\[
\boxed{6\%}
\]
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