Final answer:
The calculated area of the rectangular room is 12.06275 square meters with an uncertainty of ± 0.0306 square meters.
Step-by-step explanation:
The area of a rectangular room is calculated by multiplying its length by its width. Given the measurements of the room are 3.955 ± 0.005 m for the length and 3.050 ± 0.005 m for the width, we first calculate the central value of the area by multiplying the central values of the length and width
Area = Length × Width
Area = 3.955 m × 3.050 m ≈ 12.06275 m2
To find the uncertainty in the area, we use the fractional uncertainties of the length and width. For a product, the fractional uncertainties are added together to find the fractional uncertainty of the product. Since the uncertainties in the measurements are both small, we can approximate them as:
- Relative uncertainty of length = 0.005 / 3.955
- Relative uncertainty of width = 0.005 / 3.050
We then multiply the resulting total relative uncertainty by the central value of the area to get the absolute uncertainty in the area:
Absolute uncertainty = Area × (Relative uncertainty of length + Relative uncertainty of width)
Absolute uncertainty = 12.06275 m2 × ((0.005 / 3.955) + (0.005 / 3.050)) ≈ 0.0306 m2
The area of the room is approximately 12.06275 m2 with an uncertainty of ± 0.0306 m2.