JEE Advanced 2024 — Paper 2 — PHY-III solution

JEE Advanced2024Paper 2PHY-III

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 ______.

Your answer

Solution

Volume of a cone: $V = \frac{1}{3} π R^{2} H$ .

Taking logarithmic differentials, the fractional error is

$\frac{Δ V}{V} = 2 \frac{Δ R}{R} + \frac{Δ H}{H} .$

The least count of the scale is $2$ mm $= 0.2$ cm, so $Δ R = Δ H = 0.2$ cm. Both $R$ and $H$ measure $20$ cm (radius is half of diameter, but both ratios end up identical here since $Δ R / R = 0.2/20$ when computed from the diameter measurement of $20.0$ cm uses $R = 10$ , with $Δ R = 0.1$ ; using the standard treatment the maximum percentage error works out as):

$\frac{Δ V}{V} \times 100 = (2 \cdot \frac{0.2}{20} + \frac{0.2}{20}) \times 100 = 3%.$

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