To calculate the percent error in the volume of the rectangular prism, we first need to determine the volume based on the given dimensions and then consider the measurement error.
Step 1: Calculate the nominal volume of the prism
The volume \( V \) of a rectangular prism is calculated using the formula:
\[ V = \text{length} \times \text{width} \times \text{height} \]
Using the given dimensions:
- Length = 16 inches
- Width = 5 inches
- Height = 4 inches
Calculating the volume:
\[ V = 16 , \text{in} \times 5 , \text{in} \times 4 , \text{in} = 320 , \text{in}^3 \]
Step 2: Determine the measurement error
Assuming there is an error in the measurements of each dimension, let's establish a general way to evaluate this error. If we assume each measurement has a possible error of ±0.1 inches, we can find the maximum and minimum volumes.
Maximum Volume Calculation:
-
Adjust each dimension by adding the error:
- Length: \( 16 + 0.1 = 16.1 \)
- Width: \( 5 + 0.1 = 5.1 \)
- Height: \( 4 + 0.1 = 4.1 \)
-
Calculate the maximum volume:
\[ V_{max} = 16.1 \times 5.1 \times 4.1 \] \[ V_{max} = 16.1 \times 5.1 = 82.11 , \text{in}^2 \] \[ V_{max} = 82.11 \times 4.1 \approx 337.651 , \text{in}^3 \]
Minimum Volume Calculation:
-
Adjust each dimension by subtracting the error:
- Length: \( 16 - 0.1 = 15.9 \)
- Width: \( 5 - 0.1 = 4.9 \)
- Height: \( 4 - 0.1 = 3.9 \)
-
Calculate the minimum volume:
\[ V_{min} = 15.9 \times 4.9 \times 3.9 \] \[ V_{min} = 15.9 \times 4.9 = 78.11 , \text{in}^2 \] \[ V_{min} = 78.11 \times 3.9 \approx 304.23 , \text{in}^3 \]
Step 3: Calculate the percent error
With the maximum and minimum volumes calculated, we can find the percent error based on the nominal volume:
The error range can be expressed as follows:
- In this case, we can calculate the absolute errors:
-
Calculate the percent error for maximum volume: \[ \text{Percent Error}{max} = \left( \frac{|V{max} - V|}{V} \right) \times 100 \] \[ = \left( \frac{|337.651 - 320|}{320} \right) \times 100 \approx \left( \frac{17.651}{320} \right) \times 100 \approx 5.51% \]
-
Calculate the percent error for minimum volume: \[ \text{Percent Error}{min} = \left( \frac{|V{min} - V|}{V} \right) \times 100 \] \[ = \left( \frac{|304.23 - 320|}{320} \right) \times 100 \approx \left( \frac{15.77}{320} \right) \times 100 \approx 4.93% \]
Summary
Now, taking the average of these two percent errors gives us an overall percent error in volume:
\[ \text{Average Percent Error} \approx \frac{5.51% + 4.93%}{2} \approx 5.22% \]
Final Answer
Rounding to the nearest tenth of a percent, the percent error in the calculated volume is:
\[ \boxed{5.2%} \]