JEE Advanced 2024 Paper 2 · Q08 · Error Propagation
The dimensions of a cone are measured using a scale with a least count of 2 mm. The diameter of the base and the height are both measured to be 20.0 cm. The maximum percentage error in the determination of the volume is _____.
Reveal answer + step-by-step solution
Correct answer:3
Solution
Volume of a cone $V = \dfrac{1}{3}\pi r^2 h = \dfrac{1}{3}\pi (d/2)^2 h = \dfrac{\pi}{12} d^2 h$.
Fractional error: $\dfrac{\Delta V}{V} = 2\dfrac{\Delta d}{d} + \dfrac{\Delta h}{h}$.
Least count $= 2$ mm $= 0.2$ cm, so $\Delta d = \Delta h = 0.2$ cm. Both $d$ and $h$ are 20.0 cm.
$\dfrac{\Delta V}{V} = 2\cdot\dfrac{0.2}{20} + \dfrac{0.2}{20} = 3\cdot\dfrac{0.2}{20} = \dfrac{0.6}{20} = 0.03 = 3\%$.
Units: all in cm; relative errors are dimensionless. Answer: $3$.
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