Final answer:
To determine the area of a rectangular room with measured length and width, multiply these dimensions. The uncertainty of the area is calculated using the propagation of uncertainties, by adding the relative uncertainties of length and width and multiplying by the total area.
Step-by-step explanation:
To calculate the area of a rectangular room given the measurements, we use the formula for the area of a rectangle, which is length times width. Given that the length is 3.955 ± 0.005 m and the width is 3.050 ± 0.005 m, multiplying these together gives us the central value of the area. To find the uncertainty of the area, we use the formula for the propagation of uncertainty in multiplication, which considers the relative uncertainties of the length and width.
The central value of the area is calculated as:
Area = length × width = 3.955 m × 3.050 m = 12.06275 m²
To calculate the uncertainty, we find the relative uncertainties of the length and width, and then apply those to the area:
Relative uncertainty of length = 0.005 / 3.955
Relative uncertainty of width = 0.005 / 3.050
The combined relative uncertainty is the sum of these individual relative uncertainties, which is then multiplied by the central area to find the absolute uncertainty:
Absolute uncertainty = Area × (Relative uncertainty of length + Relative uncertainty of width)
Plugging in the numbers, we can find the uncertainty and thus state the area of the room with its uncertainty.