A solid cylindrical glass rod has length 20.0 ± 0.1 cm and diameter 5.00 ± 0.01
mm. What is the percentage uncertainty in the calculated volume of this rod?
0.1%
0.2%
0.7%
0.9%
The percentage uncertainty in the calculated volume of the solid cylindrical glass rod is 0.9%. This result comes from applying the error propagation formula to the volume V = πr²L, where uncertainties add as ΔV/V × 100% = 2(Δr/r × 100%) + (ΔL/L × 100%).
Step-by-Step Calculation
Convert diameter to consistent units: D = 5.00 mm = 0.500 cm, so ΔD = 0.01 mm = 0.001 cm. Radius r = D/2 = 0.250 cm with Δr = 0.0005 cm. Length L = 20.0 cm with ΔL = 0.1 cm.
Fractional uncertainty in length: ΔL/L × 100% = 0.1/20.0 × 100% = 0.5%. Fractional uncertainty in radius: Δr/r × 100% = 0.0005/0.250 × 100% = 0.2%. Total: 2 × 0.2% + 0.5% = 0.9%.
Option Analysis
- 0.1%: Too low; ignores dominant length uncertainty (0.5%) and doubles radius contribution.
- 0.2%: Matches only radius term; neglects length error and
r²doubling effect. - 0.7%: Close but underestimates; possibly from unit mismatch or forgetting to double radius uncertainty.
- 0.9%: Correct; precisely sums
2 × 0.2% + 0.5%.
