Question

the rectangular prism below is labeled with its measured dimensions. taking measurement error into account, what is the percent error in its calculated volume? 5 cm 14 cm 12 cm round your answer to the nearest tenth of a percent and include a percent sign (%).

Explanation:

Step1: Determine measurement error

For measurements to the nearest whole number, the absolute error is \( \frac{1}{2} = 0.5 \) cm.

Step2: Find maximum and minimum volumes

- Maximum volume: \( (14 + 0.5)(12 + 0.5)(5 + 0.5) = 14.5 \times 12.5 \times 5.5 \)
\( 14.5 \times 12.5 = 181.25 \), \( 181.25 \times 5.5 = 996.875 \) \( \text{cm}^3 \)
- Minimum volume: \( (14 - 0.5)(12 - 0.5)(5 - 0.5) = 13.5 \times 11.5 \times 4.5 \)
\( 13.5 \times 11.5 = 155.25 \), \( 155.25 \times 4.5 = 698.625 \) \( \text{cm}^3 \)
- Measured volume: \( 14 \times 12 \times 5 = 840 \) \( \text{cm}^3 \)

Step3: Calculate absolute error

Absolute error is the maximum of \( |996.875 - 840| \) and \( |698.625 - 840| \)
\( |996.875 - 840| = 156.875 \), \( |698.625 - 840| = 141.375 \). So absolute error is \( 156.875 \)

Step4: Calculate percent error

Percent error formula: \( \frac{\text{Absolute Error}}{\text{Measured Value}} \times 100\% \)
\( \frac{156.875}{840} \times 100\% \approx 18.7\% \) (Wait, no, correction: Wait, actually, for measurement error in each dimension, the error in volume is approximated by the sum of relative errors times the measured volume. Wait, maybe a better approach: The relative error in each dimension is \( \frac{0.5}{14} \), \( \frac{0.5}{12} \), \( \frac{0.5}{5} \). Then the relative error in volume is approximately the sum of relative errors (for multiplication, relative errors add). Let's recalculate:

Relative error for length: \( \frac{0.5}{14} \approx 0.0357 \)
Relative error for width: \( \frac{0.5}{12} \approx 0.0417 \)
Relative error for height: \( \frac{0.5}{5} = 0.1 \)
Total relative error ≈ \( 0.0357 + 0.0417 + 0.1 = 0.1774 \)
Percent error ≈ \( 0.1774 \times 100\% \approx 17.7\% \)? Wait, no, my first method was wrong because when calculating maximum volume, I added 0.5 to each, but the error is ±0.5, so the absolute error in volume is \( \Delta V = V_{\text{max}} - V_{\text{measured}} \) or \( V_{\text{measured}} - V_{\text{min}} \), whichever is larger.

Wait, let's recalculate \( V_{\text{max}} = (14 + 0.5)(12 + 0.5)(5 + 0.5) = 14.5 \times 12.5 \times 5.5 \)
14.5*12.5 = 181.25; 181.25*5.5 = 996.875
\( V_{\text{min}} = (14 - 0.5)(12 - 0.5)(5 - 0.5) = 13.5 \times 11.5 \times 4.5 \)
13.5*11.5 = 155.25; 155.25*4.5 = 698.625
Measured V = 14*12*5 = 840
So \( \Delta V_{\text{max}} = 996.875 - 840 = 156.875 \)
\( \Delta V_{\text{min}} = 840 - 698.625 = 141.375 \)
The absolute error is the maximum of these two, which is 156.875
Then percent error is \( \frac{156.875}{840} \times 100\% \approx 18.7\% \)? But that seems high. Wait, maybe the problem assumes that the measurement error is ±0.5 cm (since the dimensions are whole numbers, so the precision is to the nearest cm, so the error is ±0.5 cm). Then the formula for percent error in volume when each dimension has error \( \Delta x, \Delta y, \Delta z \) is \( \frac{(x + \Delta x)(y + \Delta y)(z + \Delta z) - xyz}{xyz} \times 100\% \) (taking the maximum error, i.e., all errors positive or all negative, whichever gives larger absolute value).

So \( (14 + 0.5)(12 + 0.5)(5 + 0.5) - 14*12*5 = 14.5*12.5*5.5 - 840 \)
14.5*12.5 = 181.25; 181.25*5.5 = 996.875; 996.875 - 840 = 156.875
Then percent error is \( \frac{156.875}{840} \times 100\% \approx 18.7\% \)? But let's check with relative errors. The relative error in x is 0.5/14 ≈ 0.0357, y: 0.5/12 ≈ 0.0417, z: 0.5/5 = 0.1. The relative error in volume is approximately the sum of relative errors (for multiplication, the relative error of the product is approximately the sum of relative errors of the factors). So 0.0357 + 0.0417 + 0.1 = 0.1774, so 17.74%, which is close to 17.7% or 18.7%? Wait, my first calculation with absolute error gives 156.875 / 840 ≈ 0.186759, so 18.7% when rounded to the nearest tenth. Wait, maybe the problem expects this method. Let's do the division: 156.875 ÷ 840 = 0.186759... Multiply by 100: 18.6759..., which rounds to 18.7%.

Wait, but let's confirm. The measured volume is 14*12*5 = 840. The maximum possible volume is (14.5)(12.5)(5.5) = 996.875. The difference is 996.875 - 840 = 156.875. So percent error is (156.875 / 840) * 100 ≈ 18.7% (rounded to the nearest tenth).

Answer:

18.7%