To calculate the volume of a rectangular box and its uncertainty, we need to follow these steps:
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Calculate the Volume: The volume \( V \) of a rectangular box is given by the formula: \[ V = l \times w \times h \] where \( l \), \( w \), and \( h \) are the lengths of the sides of the box.
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Given Measurements:
- Length \( l = 1.80 \pm 0.01 \) cm
- Width \( w = 2.05 \pm 0.02 \) cm
- Height \( h = 3.10 \pm 0.01 \) cm
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Calculate the Volume: \[ V = 1.80 , \text{cm} \times 2.05 , \text{cm} \times 3.10 , \text{cm} \]
Performing the multiplication: \[ V = 1.80 \times 2.05 \times 3.10 \approx 11.3 , \text{cm}^3 \]
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Calculate the Uncertainty: The uncertainty in volume can be calculated using the formula for the uncertainty of multiplication: \[ \frac{\Delta V}{V} = \sqrt{\left(\frac{\Delta l}{l}\right)^2 + \left(\frac{\Delta w}{w}\right)^2 + \left(\frac{\Delta h}{h}\right)^2} \]
Where \( \Delta l \), \( \Delta w \), and \( \Delta h \) are the uncertainties in the lengths.
Substituting the values:
- \( \Delta l = 0.01 , \text{cm} \)
- \( \Delta w = 0.02 , \text{cm} \)
- \( \Delta h = 0.01 , \text{cm} \)
We first calculate the relative uncertainties: \[ \frac{\Delta l}{l} = \frac{0.01}{1.80} \approx 0.00556 \] \[ \frac{\Delta w}{w} = \frac{0.02}{2.05} \approx 0.00976 \] \[ \frac{\Delta h}{h} = \frac{0.01}{3.10} \approx 0.00323 \]
Now substitute into the total relative uncertainty formula: \[ \frac{\Delta V}{V} = \sqrt{(0.00556)^2 + (0.00976)^2 + (0.00323)^2} \] \[ \frac{\Delta V}{V} \approx \sqrt{0.0000309136 + 0.0000952976 + 0.0000104289} \] \[ \frac{\Delta V}{V} \approx \sqrt{0.0001366401} \approx 0.0117 \]
Now, calculate \( \Delta V \): \[ \Delta V \approx V \times 0.0117 \approx 11.3 \times 0.0117 \approx 0.132 \]
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Final Result: Rounding \( \Delta V \) gives approximately \( 0.13 , \text{cm}^3 \). Thus, the volume of the box with uncertainty is: \[ V \approx 11.3 \pm 0.13 , \text{cm}^3 \]
Therefore, the closest answer would be:
- \(11.3 \pm 0.13 , \text{cm}^3\).
In your options, none aligns directly with this calculation. The initial volume estimate provided in the options as \( 11.4 \pm 0.21 , \text{cm}^3 \) could be considered as a rounding that is close to the values calculated if we adjust our assumptions about error margins, making \( 11.4 \pm 0.21 , \text{cm}^3 \) the most reasonable option.