Question:
The density of a solid metal sphere is determined by measuring its mass and its diameter. The maximum error in the density of the sphere is \(\left(\frac{x}{100}\right)\%\). If the relative errors in measuring the mass and the diameter are 6.0% and 1.5% respectively, the value of x is_______.
The density of a solid metal sphere is determined by measuring its mass and its diameter. The maximum error in the density of the sphere is \(\left(\frac{x}{100}\right)\%\). If the relative errors in measuring the mass and the diameter are 6.0% and 1.5% respectively, the value of x is_______.
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Density \(\rho=\frac{M}{\frac{\pi}{6}d^3}\); the maximum fractional error is \(\frac{\Delta M}{M}+3\frac{\Delta d}{d}\).
Updated On: Nov 5, 2026
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Correct Answer: 1050
Solution and Explanation
\(\rho=\frac{M}{\frac{4}{3}\pi (d/2)^3}=\frac{6M}{\pi d^3}\).
Maximum relative error: \(\frac{\Delta\rho}{\rho}=\frac{\Delta M}{M}+3\frac{\Delta d}{d}\).
\(=6.0\%+3\times1.5\%=6.0\%+4.5\%=10.5\%\).
Given \(\frac{x}{100}=10.5\), so \(x=1050\).
Answer: 1050
Maximum relative error: \(\frac{\Delta\rho}{\rho}=\frac{\Delta M}{M}+3\frac{\Delta d}{d}\).
\(=6.0\%+3\times1.5\%=6.0\%+4.5\%=10.5\%\).
Given \(\frac{x}{100}=10.5\), so \(x=1050\).
Answer: 1050
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