Question

. The length and width of a rectangular room are measured to be 3.955 ± 0.005 m and

3.050 ± 0.005 m. Calculate (a) the relative and percentage errors in measuring the
length and width; (b) the area of the room and its uncertainty in square met

Answer 1

(a)

Step-by-step explanation:

The relative error is the ratio of the uncertainty (or error) to the measured value. The percentage error is the relative error multiplied by 100%.

For the length:

Relative error = 0.005 m / 3.955 m = 0.00126

Percentage error = 0.00126 * 100% = 0.126%

For the width:

Relative error = 0.005 m / 3.050 m = 0.00164

Percentage error = 0.00164 * 100% = 0.164%

(b)

The area of the room is given by:

Area = length × width

We can use the formula for the propagation of uncertainty to find the uncertainty in the area, which is given by:

σ_Area = sqrt((σ_length/length)^2 + (σ_width/width)^2) × Area

where σ_length and σ_width are the uncertainties in the length and width measurements, respectively.

Substituting the values:

σ_Area = sqrt((0.005/3.955)^2 + (0.005/3.050)^2) × (3.955 × 3.050) m^2

σ_Area = 0.014 m^2

Therefore, the area of the room is (3.955 × 3.050) m^2, with an uncertainty of ±0.014 m^2.