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Error in Real Problems

A worked, multi-step example combining bounds and appropriate precision on a realistic measurement problem.

A Full Worked Problem

Example 1 — bounds for a garden's area

A rectangular garden has length 8.4 m and width 5.2 m, both measured to 1 decimal place. Find the upper and lower bounds of its area. 1 Bounds of the length: 8.35 m to 8.45 m 2 Bounds of the width: 5.15 m to 5.25 m 3 Maximum area = max \times max = 8.45 \times 5.25 = 44.3625 4 Minimum area = min \times min = 8.35 \times 5.15 = 43.0025 Answer: Maximum \approx 44.36 m^{2}, Minimum \approx 43.00 m^{2}

Example 2 — reporting the area with appropriate precision

Using the same garden, report a single sensible value for the area, using appropriate precision. 1 The raw calculation: 8.4 \times 5.2 = 43.68 2 Both inputs have 2 significant figures — match that in the final answer 3 Round 43.68 to 2 significant figures Answer: \approx 44 m^{2} — note this matches the range 43.00 to 44.36 m^{2} found in Example 1, so it is a fair, honest summary of the real uncertainty.

Practice

A rectangle has length 6.5 m and width 3.2 m, both to 1 decimal place. Find the upper bound of its area, to 2 decimal places.

Your answer

A Full Worked Problem

Example 1 — bounds for a garden's area

A rectangular garden has length 8.4 m and width 5.2 m, both measured to 1 decimal place. Find the upper and lower bounds of its area. 1 Bounds of the length: 8.35 m to 8.45 m 2 Bounds of the width: 5.15 m to 5.25 m 3 Maximum area = max \times max = 8.45 \times 5.25 = 44.3625 4 Minimum area = min \times min = 8.35 \times 5.15 = 43.0025 Answer: Maximum \approx 44.36 m^{2}, Minimum \approx 43.00 m^{2}

Example 2 — reporting the area with appropriate precision

Using the same garden, report a single sensible value for the area, using appropriate precision. 1 The raw calculation: 8.4 \times 5.2 = 43.68 2 Both inputs have 2 significant figures — match that in the final answer 3 Round 43.68 to 2 significant figures Answer: \approx 44 m^{2} — note this matches the range 43.00 to 44.36 m^{2} found in Example 1, so it is a fair, honest summary of the real uncertainty.

Practice

A rectangle has length 6.5 m and width 3.2 m, both to 1 decimal place. Find the upper bound of its area, to 2 decimal places.

Your answer