College Physics Problem 1.27


The length and width of a rectangular room are measured to be 3.955 ±0.005 m and 3.050 ± 0.005 m . Calculate the area of the room and its uncertainty in square meters.


Solution:

The average area of the room is

\displaystyle A=l\times w

\displaystyle A=3.955\:m\times 3.050\:m

\displaystyle A=12.06\:m^2

Compute for the percent uncertainties of each dimension.

\displaystyle \%\:unc_{width}=\frac{0.005\:m}{3.050\:m}\times 100\%=0.1639\%

\displaystyle \%\:unc_{length}=\frac{0.005\:m}{3.955\:m}\times 100\%=0.1264\:\%

The percent uncertainty in the area is the combined effect of the uncertainties of the length and width.

\displaystyle \%\:unc_{area}=0.1639\%+0.1264\%=0.2903\%

The uncertainty in the area is

\displaystyle \delta _{area}=\frac{0.2903\:\%}{100\:\%}\times 12.06\:m^2=0.035\:m^2

Therefore, the area is

\displaystyle A=12.06\pm 0.035\:m^2