Question

the rectangle below is labeled with its measured dimensions. taking measurement error into account, what is the percent error in its calculated area?
20 mm
13 mm
round your answer to the nearest tenth of a percent and include a percent sign (%).
submit

Explanation:

Step1: Determine measurement error

For a measured value, the absolute error is typically half the smallest unit of measurement. Assuming the measurements are to the nearest millimeter, the absolute error for each dimension is \( \frac{1}{2} = 0.5 \) mm.

Step2: Find maximum and minimum areas

• Maximum area: Use the maximum possible dimensions (\( 20 + 0.5 = 20.5 \) mm and \( 13 + 0.5 = 13.5 \) mm).

\( A_{\text{max}} = 20.5 \times 13.5 = 276.75 \) mm².

• Minimum area: Use the minimum possible dimensions (\( 20 - 0.5 = 19.5 \) mm and \( 13 - 0.5 = 12.5 \) mm).

\( A_{\text{min}} = 19.5 \times 12.5 = 243.75 \) mm².

Step3: Calculate measured area

The measured area is \( A_{\text{measured}} = 20 \times 13 = 260 \) mm².

Step4: Find absolute error in area

The absolute error is the maximum of \( |A_{\text{max}} - A_{\text{measured}}| \) and \( |A_{\text{min}} - A_{\text{measured}}| \):
\( |276.75 - 260| = 16.75 \), \( |243.75 - 260| = 16.25 \).
So, absolute error \( = 16.75 \) mm².

Step5: Calculate percent error

Percent error formula: \( \text{Percent Error} = \frac{\text{Absolute Error}}{\text{Measured Value}} \times 100\% \)
\( \text{Percent Error} = \frac{16.75}{260} \times 100\% \approx 6.4\% \)

Answer:

6.4%