Question

12. (10 points) The dimensions of a room have been measured as 10.54 + 0.05m and 25.72+0.07m. Find the uncertainty of that area. Show your work.. (-10 points is significant figures are incorrect.)

Answer 1

Area of a rectangular section of dimensions x and y is given by

  $A=xy$
And the uncertainty is given by

  $\frac{\Delta A}{A}=\bigg|\frac{\Delta x}{x}\bigg|+\bigg|\frac{\Delta y}{y}\bigg|$
And so, we have

$\Rightarrow \Delta A=A\bigg[\bigg|\frac{\Delta x}{x}\bigg|+\bigg|\frac{\Delta y}{y}\bigg|\bigg]$
So, given the values,

  $x\pm \Delta x=(10.54\pm 0.05)~m$
And
$y\pm \Delta y=(25.72\pm 0.07)~m$
So, the area is given by

$A=xy=10.54\times 25.72=271.0888~m^2$
And so, the uncertainty in the area is given by
$\Rightarrow \Delta A=271.0888\times \bigg[\bigg|\frac{0.05}{10.54}\bigg|+\bigg|\frac{0.07}{25.72}\bigg|\bigg]$
  $\Rightarrow \Delta A=2.0238~m^2$
And so, we have the area with the uncertainty as
  $A\pm \Delta A=(271.0888\pm 2.0238)~m^2$