The length and width of a rectangular room are measured to be 3.955±0.005m3.955±0.005m and 3.050±0.005m3.050±0.005m . Calculate the area of the room and its uncertainty in square meters.
Question
The length and width of a rectangular room are measured to be 3.955±0.005m3.955±0.005m and 3.050±0.005m3.050±0.005m . Calculate the area of the room and its uncertainty in square meters.
Solution
The area of a rectangle is calculated by multiplying the length and the width.
So, the area of the room is 3.955m * 3.050m = 12.06225 m².
The uncertainty in a product or quotient is given by the sum of the relative uncertainties of each measurement.
The relative uncertainty of each measurement is the absolute uncertainty divided by the measurement itself.
So, the relative uncertainty of the length is 0.005m / 3.955m = 0.0012642225031605564.
The relative uncertainty of the width is 0.005m / 3.050m = 0.001639344262295082.
Adding these together gives a total relative uncertainty of 0.0029035667654556386.
Multiplying this by the area gives the uncertainty in the area: 0.0029035667654556386 * 12.06225 m² = 0.035 m².
So, the area of the room is 12.06225 ± 0.035 m².
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