To calculate the volume of the rectangular box, we use the formula for the volume of a rectangular box:

$V = l \cdot w \cdot h$

where:

• $l$ is the length,
• $w$ is the width,
• $h$ is the height.

Step 1: Calculate the nominal volume

Nominal dimensions: $l = 1.80 \, \text{cm}, \, w = 2.05 \, \text{cm}, \, h = 3.1 \, \text{cm}$

Nominal volume: $V = 1.80 \cdot 2.05 \cdot 3.1 = 11.439 \, \text{cm}^3$

Step 2: Calculate the relative uncertainties

Relative uncertainty is given by: $\text{Relative uncertainty} = \frac{\text{Absolute uncertainty}}{\text{Nominal value}}$

For each dimension: $\text{Relative uncertainty of } l = \frac{0.01}{1.80} \approx 0.00556$ $\text{Relative uncertainty of } w = \frac{0.02}{2.05} \approx 0.00976$ $\text{Relative uncertainty of } h = \frac{0.1}{3.1} \approx 0.03226$

Step 3: Calculate the total relative uncertainty

The total relative uncertainty for the volume is the sum of the relative uncertainties of all dimensions: $\text{Total relative uncertainty} = 0.00556 + 0.00976 + 0.03226 = 0.04758$

Step 4: Calculate the absolute uncertainty in volume

The absolute uncertainty in volume is given by: $\text{Absolute uncertainty in } V = \text{Nominal volume} \cdot \text{Total relative uncertainty}$ $\text{Absolute uncertainty in } V = 11.439 \cdot 0.04758 \approx 0.5445 \, \text{cm}^3$

Step 5: Final result with uncertainty

The volume of the box is: $V = 11.44 \pm 0.54 \, \text{cm}^3$

Let me know if you need further clarifications!

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Related Questions

1. How is relative uncertainty calculated, and why is it important?
2. Can you explain how uncertainties propagate in multiplication or division?
3. How does increasing the accuracy of measurement tools reduce the uncertainty in the final result?
4. What is the impact of one dimension having a much higher uncertainty than the others?
5. How can this method be applied to irregular shapes?

Tip:

Always keep your measurements consistent in terms of significant figures when presenting uncertainties.