Question

or
The rectangle below is labeled with its measured dimensions. Taking measurement error into account, what is the percent error in its calculated area?

18 ft
4 ft


Round your answer to the nearest tenth of a percent and include a percent sign (%).

Answer 1

To find the percent error in the calculated area of the rectangle, we first need to calculate the area and then consider the measurement errors.

1. **Calculate the area:**
The formula for the area \( A \) of a rectangle is:
\[
A = \text{length} \times \text{width}
\]
Given the dimensions are 18 ft and 4 ft, the area is:
\[
A = 18 \, \text{ft} \times 4 \, \text{ft} = 72 \, \text{ft}^2
\]

2. **Consider measurement error:**
We need to know the measurement errors for both dimensions. For this example, let’s assume there is a measurement error of ±0.5 ft for each dimension.

- For length (18 ft), the possible measurements including error can vary from:
\[
17.5 \, \text{ft} \, \text{to} \, 18.5 \, \text{ft}
\]
- For width (4 ft), the possible measurements can vary from:
\[
3.5 \, \text{ft} \, \text{to} \, 4.5 \, \text{ft}
\]

3. **Calculate the minimum and maximum areas:**
- **Minimum area** (using minimum dimensions):
\[
A_{\text{min}} = 17.5 \, \text{ft} \times 3.5 \, \text{ft} = 61.25 \, \text{ft}^2
\]

- **Maximum area** (using maximum dimensions):
\[
A_{\text{max}} = 18.5 \, \text{ft} \times 4.5 \, \text{ft} = 83.25 \, \text{ft}^2
\]

4. **Calculate percent error:**
Percent error can be calculated using the formula:
\[
\text{Percent Error} = \left( \frac{\text{Measured Value} - \text{True Value}}{\text{True Value}} \right) \times 100\%
\]

Using the calculated areas:
- For minimum area:
\[
\text{Percent Error}_{\text{min}} = \left( \frac{61.25 - 72}{72} \right) \times 100\% \approx -14.5\%
\]

- For maximum area:
\[
\text{Percent Error}_{\text{max}} = \left( \frac{83.25 - 72}{72} \right) \times 100\% \approx 15.6\%
\]

5. **Finding overall percent error:**
The percent error in the calculated area will consider both the maximum and minimum possible areas. The average of the absolute percent errors gives a general sense of the overall measurement error. Let’s calculate both:

- Average absolute error:
\[
\text{Average Absolute Error} = \frac{|-14.5| + |15.6|}{2} = \frac{14.5 + 15.6}{2} = 15.05\%
\]

6. **Round the answer:**
Therefore, the percent error in the calculated area, rounded to the nearest tenth of a percent, is:
\[
\boxed{15.1\%}
\]