Uncertainty propagation for circle area

Steps:

• Calculate radius: r = d/2 = 7.0/2 = 3.5 mm, with δr = δd/2 = 0.1/2 = 0.05 mm.
• Compute area: A = π r² ≈ 3.14 × (3.5)² = 3.14 × 12.25 ≈ 38.5 mm².
• Determine uncertainty: δA = 2 π r δr ≈ 2 × 3.14 × 3.5 × 0.05 ≈ 1.1 mm².
• Report as 38.5 ± 1.1 mm² (option B).

Why B is correct:

• Uses the formula for uncertainty in area, δA = 2 π r δr, from differentiation of A = π r².

Why the others are wrong:

• A: Incorrect uncertainty; uses relative error of radius instead of twice that for area (δA/A = 2 δr/r).
• C: Wrong area; calculates π d² ≈ 154 mm², forgetting to divide by 4.
• D: Wrong area and uncertainty; same error as C, with exaggerated δA ≈ 2 × (π d δd) ≈ 4 mm².

Final answer: B