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Difficulty: MediumErrors in Measurement and Significant Figures

The length and width of a rectangular metal sheet are measured as (8.0±0.2) cm(8.0 \pm 0.2)\text{ cm} and (5.0±0.1) cm(5.0 \pm 0.1)\text{ cm}, respectively. What is the percentage error in the calculated area of the metal sheet?

Answer: 4.5 %

Answer

The percentage error in the calculated area of the metal sheet is 4.5%4.5\%.
For a calculated quantity involving multiplication (A=L×WA = L \times W), the total relative error is the sum of the relative errors of the individual measurements. The fractional error in length is 0.28.0=0.025\frac{0.2}{8.0} = 0.025 (2.5%2.5\%) and in width is 0.15.0=0.020\frac{0.1}{5.0} = 0.020 (2.0%2.0\%). Summing these gives 0.0450.045, which is equivalent to 4.5%4.5\%.

Step-by-Step Solution

1
Determine the fractional error in the length measurement
0.2 cm8.0 cm=0.025\frac{0.2\text{ cm}}{8.0\text{ cm}} = 0.025
Fractional error is given by the ratio of absolute error to the measured value.
2
Determine the fractional error in the width measurement
0.1 cm5.0 cm=0.020\frac{0.1\text{ cm}}{5.0\text{ cm}} = 0.020
Fractional error is calculated as the absolute uncertainty divided by the measured dimension.
3
Combine the fractional errors for the calculated area
\frac{\Delta A}{A} = 0.025 + 0.020 = 0.045
For quantities multiplied together (A=L×WA = L \times W), individual fractional errors sum to give the total fractional error.
4
Convert the fractional error to percentage error
0.045×100%=4.5%0.045 \times 100\% = 4.5\%
Multiplying the relative error by 100 yields the percentage error.

Key Concept

Error Propagation in Products
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Topics

Errors in Measurement and Significant Figures

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