Question:
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 \(20.0 \, \text{cm}\). The maximum percentage error in the determination of the volume is \_\_\_\_.
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 \(20.0 \, \text{cm}\). The maximum percentage error in the determination of the volume is \_\_\_\_.
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When calculating percentage errors, sum the relative errors contributed by each term in the formula, multiplied by their respective powers.
Updated On: Jan 20, 2025
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Solution and Explanation
The volume of the cone is given by:
\[
V = \frac{1}{3} \pi R^2 H.
\]
The relative error in volume is:
\[
\frac{dV}{V} = 2 \frac{dR}{R} + \frac{dH}{H}.
\]
The percentage error in measuring the volume:
\[
\% \text{error in measuring volume} = \left[ 2 \cdot \frac{0.2}{20} + \frac{0.2}{20} \right] \cdot 100.
\]
Simplifying:
\[
\% \text{error in measuring volume} = \left[ 2 \cdot 0.01 + 0.01 \right] \cdot 100 = 3.
\]
Thus, the maximum percentage error in the determination of the volume is \(3\%\).
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