JEE Mains · Physics · STD 11 - Motion in plane
- JEE Mains 2026
- Medium
- MCQ
The percentage error in the calculated volume of a sphere, if there is \(2\%\) error in its diameter measurement, is __________.
Pick an option to check your answer.
Answer & Solution
Correct Answer
(C) \(6\)
Step-by-step Solution
Detailed explanation
The volume of a sphere in terms of its diameter \(D\) is given by \(V = \dfrac{\pi}{6}D^3\)
Taking the natural logarithm on both sides, we get \(\ln V = \ln\left(\dfrac{\pi}{6}\right) + 3 \ln D\)
Differentiating for small errors, we obtain \(\dfrac{\Delta V}{V} = 3 \dfrac{\Delta D}{D}\)
Multiplying by \(100\) to find the percentage error, \(\dfrac{\Delta V}{V} \times 100 = 3 \left(\dfrac{\Delta D}{D} \times 100\right)\)
Given that the percentage error in the diameter is \(2\%\), we have \(\dfrac{\Delta D}{D} \times 100 = 2\)
Therefore, the percentage error in the volume is \(3 \times 2 = 6\%\)
Answer: \(6\)
Taking the natural logarithm on both sides, we get \(\ln V = \ln\left(\dfrac{\pi}{6}\right) + 3 \ln D\)
Differentiating for small errors, we obtain \(\dfrac{\Delta V}{V} = 3 \dfrac{\Delta D}{D}\)
Multiplying by \(100\) to find the percentage error, \(\dfrac{\Delta V}{V} \times 100 = 3 \left(\dfrac{\Delta D}{D} \times 100\right)\)
Given that the percentage error in the diameter is \(2\%\), we have \(\dfrac{\Delta D}{D} \times 100 = 2\)
Therefore, the percentage error in the volume is \(3 \times 2 = 6\%\)
Answer: \(6\)
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