Uncertainty Propagation in Density Calculation

Steps:

• Density ρ = m / V, where V = (4/3)π(r)^3 and r = d/2, so V ∝ d^3.
• Relative uncertainty in V is 3 × (δd/d) = 3 × (0.04/1.87) ≈ 3 × 0.0214 = 0.0642.
• Relative uncertainty in ρ is δm/m + δV/V = (0.1/32.5) + 0.0642 ≈ 0.0031 + 0.0642 = 0.0673.
• Percentage uncertainty = 100 × 0.0673 ≈ 6.7%, which rounds to 7%.

Why D is correct:

• The formula for density uncertainty δρ/ρ = δm/m + 3(δd/d) accounts for the cubic dependence of volume on diameter, yielding ~7%.

Why the others are wrong:

• A: Neglects all uncertainties, just the measured value.
• B: Uses only δd/d without the factor of 3 for volume.
• C: Applies 3(δd/d) but omits δm/m, giving ~6.4%.

Final answer: D