Question:
If error in diameter of a sphere is 2%, then find error in volume of the sphere.
If error in diameter of a sphere is 2%, then find error in volume of the sphere.
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The error in the volume of a sphere is three times the error in its radius, which in turn is half the error in its diameter.
Updated On: Apr 7, 2026
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Correct Answer: 6
Solution and Explanation
Step 1: Understanding the relation between volume and diameter.
The volume \( V \) of a sphere is given by the formula: \[ V = \frac{4}{3} \pi r^3 \] where \( r \) is the radius of the sphere. The diameter \( d \) is related to the radius as: \[ d = 2r \] Thus, the error in the diameter also affects the error in the radius.
Step 2: Relating the error in diameter to the error in volume.
The error in volume is related to the error in radius by the following relation: \[ \frac{\Delta V}{V} = 3 \frac{\Delta r}{r} \] Since \( \Delta r = \frac{\Delta d}{2} \), the error in the radius is half the error in the diameter. Given that the error in diameter is 2%, the error in radius is: \[ \frac{\Delta r}{r} = \frac{2}{2} = 2\% \] Now, the error in volume becomes: \[ \frac{\Delta V}{V} = 3 \times 2\% = 6\% \]
Step 3: Conclusion.
Therefore, the error in the volume of the sphere is 6%.
The volume \( V \) of a sphere is given by the formula: \[ V = \frac{4}{3} \pi r^3 \] where \( r \) is the radius of the sphere. The diameter \( d \) is related to the radius as: \[ d = 2r \] Thus, the error in the diameter also affects the error in the radius.
Step 2: Relating the error in diameter to the error in volume.
The error in volume is related to the error in radius by the following relation: \[ \frac{\Delta V}{V} = 3 \frac{\Delta r}{r} \] Since \( \Delta r = \frac{\Delta d}{2} \), the error in the radius is half the error in the diameter. Given that the error in diameter is 2%, the error in radius is: \[ \frac{\Delta r}{r} = \frac{2}{2} = 2\% \] Now, the error in volume becomes: \[ \frac{\Delta V}{V} = 3 \times 2\% = 6\% \]
Step 3: Conclusion.
Therefore, the error in the volume of the sphere is 6%.
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