A cylindrical sample has mass (44.4 ± 0.2) g, diameter (2.40 ± 0.05) cm and length (5.00 ± 0.05) cm. Use π = 3.142. (a) Distinguish accuracy from precision. [2] (b) Calculate the cylinder volume and density. [4] (c) Estimate the percentage uncertainty in the density by adding percentage uncertainties; remember that diameter is squared. [4] (d) Reference densities are aluminium 2.70 g/cm³ and magnesium 1.74 g/cm³. Judge whether density alone securely identifies the sample. [2]

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The volume is 22.62 cm³ and the density is 1.96 g/cm³. The estimated percentage uncertainty is 5.62%, or about ±0.11 g/cm³, so the interval does not match either reference closely enough to identify the alloy securely. (a) Precision describes how closely repeated readings agree; accuracy describes closeness to an accepted value. A set can be precise yet inaccurate if every reading has the same calibration offset. (b) Radius = 2.40/2 = 1.20 cm. Volume V = πr²L = 3.142 × 1.20² × 5.00 = 22.62 cm³, with units carried through the calculation. (b) Density ρ = m/V = 44.4/22.62 = 1.96 g/cm³ to three significant figures, matching the precision of the supplied measurements. (c) Mass uncertainty = 0.2/44.4 × 100 = 0.45%. Volume uncertainty ≈ 2(0.05/2.40 × 100) + 0.05/5.00 × 100 = 5.17%. (c) Density is a quotient, so the estimated percentage uncertainty is 0.45% + 5.17% = 5.62%. Its absolute uncertainty is 0.0562 × 1.96 ≈ 0.11 g/cm³. (d) The plausible range is approximately 1.85–2.07 g/cm³. It excludes the two pure reference values, suggesting an alloy or a systematic measurement problem; density alone cannot determine composition.

Can the density identify an unknown alloy?

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