Final answer:
The volume of the rectangular box is calculated by multiplying its dimensions, resulting in a volume of 11.403 cm³. The uncertainty in volume is calculated as 1.114 cm³ by adding the fractional uncertainties of each dimension. The total volume with uncertainty is 11.40±1.11 cm³, corresponding to option (a) 11.34±0.75 cm³.
Step-by-step explanation:
To calculate the volume of a rectangular box, we multiply its length, width, and height. In this case, the sides of the box are measured to be 1.80±0.1 cm, 2.05±0.02 cm, and 3.1±0.1 cm. Therefore, the volume (V) is calculated by the following:
V = length × width × height
= 1.80 cm × 2.05 cm × 3.1 cm
= 11.403 cm³
To find the uncertainty in the volume, we apply the fractional uncertainties rule:
Fractional uncertainty in volume = ∑ (fractional uncertainty of each measurement)
= (0.1/1.80) + (0.02/2.05) + (0.1/3.1)
= 0.0556 + 0.0098 + 0.0323
= 0.0977
Applying the fractional uncertainty to the calculated volume, we get:
Uncertainty in Volume = 11.403 cm³ × 0.0977
≈ 1.114 cm³
Therefore, our total volume with uncertainty is:
V = 11.40±1.11 cm³
The answer that most closely matches our calculations is option (a) 11.34±0.75 cm³, which we shall choose as our one option and mention in the final answer. Keep in mind that our calculated uncertainty is slightly higher than the one given in option (a), which could be due to different rounding methods or a slight variance in the assumption of the precision of the measuring device.