To calculate the volume of a rectangular box, we use the formula:
\[ V = L \times W \times H \]
where \( L \) is the length, \( W \) is the width, and \( H \) is the height of the box.
Given the measurements:
- Length \( L = 1.80 , \text{cm} \) with an uncertainty of \( \Delta L = 0.01 , \text{cm} \)
- Width \( W = 2.05 , \text{cm} \) with an uncertainty of \( \Delta W = 0.02 , \text{cm} \)
- Height \( H = 3.10 , \text{cm} \) with an uncertainty of \( \Delta H = 0.01 , \text{cm} \)
First, we calculate the volume:
\[ V = 1.80 \times 2.05 \times 3.10 \]
Calculating step by step:
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Calculate \( 1.80 \times 2.05 \): \[ 1.80 \times 2.05 = 3.69 \]
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Calculate \( 3.69 \times 3.10 \): \[ 3.69 \times 3.10 = 11.439 \]
Thus, the volume of the box is:
\[ V \approx 11.439 , \text{cm}^3 \]
Next, we calculate the uncertainty in volume using the formula for propagation of uncertainty for multiplication:
\[ \frac{\Delta V}{V} = \sqrt{\left(\frac{\Delta L}{L}\right)^2 + \left(\frac{\Delta W}{W}\right)^2 + \left(\frac{\Delta H}{H}\right)^2} \]
Calculating each relative uncertainty:
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For length \( L \): \[ \frac{\Delta L}{L} = \frac{0.01}{1.80} \approx 0.00556 \]
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For width \( W \): \[ \frac{\Delta W}{W} = \frac{0.02}{2.05} \approx 0.00976 \]
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For height \( H \): \[ \frac{\Delta H}{H} = \frac{0.01}{3.10} \approx 0.00323 \]
Now we can compute the total relative uncertainty in volume:
\[ \frac{\Delta V}{V} = \sqrt{(0.00556)^2 + (0.00976)^2 + (0.00323)^2} \]
Calculating each term:
- \( (0.00556)^2 \approx 0.0000309136 \)
- \( (0.00976)^2 \approx 0.0000956976 \)
- \( (0.00323)^2 \approx 0.0000104329 \)
Now summing these:
\[ \frac{\Delta V}{V} = \sqrt{0.0000309136 + 0.0000956976 + 0.0000104329} \approx \sqrt{0.0001360441} \approx 0.01165 \]
Multiply \( 0.01165 \) by the volume \( V \) to find \( \Delta V \):
\[ \Delta V = V \times 0.01165 \approx 11.439 \times 0.01165 \approx 0.1334 \]
Thus, rounding to two decimal places, the uncertainty in volume is approximately \( 0.13 , \text{cm}^3 \).
Finally, we report the volume with its uncertainty:
\[ V = 11.44 \pm 0.13 , \text{cm}^3 \]
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