Calculating uncertainty from significant figures of a value

Question

A question in Giancoli's Physics for Scientists and Engineers (2. ed) has me confused. Here it is (Ch 1, Problem 3):

What is the area, and its approximate uncertainty, of a circle of radius $2.7 \times 10^4$ cm?

I got the correct answer of $2.3 \times 10^9 \text{ cm}^2$, but the uncertainty provided in the answer was $0.2 \times 10^9 \text{ cm}^2$. How was this uncertainty calculated? As far as I can tell, it is not possible to determine the uncertainty of a measure just from its value, because I have no idea with what device/technology the radius was measured. All we can know is that the doubtful figure is in the $10^3$ position, and therefore the uncertainty in the area will be $x \times 10^9$, but $x$ could be anything? Why is $x = 0.2$?

Answer 1

Most likely, the authors assume Gauss error propagation in which the error on a function $f(x)$ of a variable $x$ is calculated as $$ \Delta f = \frac{\partial f}{\partial x} \Delta x~.$$ In your case, $f(x) = \pi x^2$ and $\frac{\partial f}{\partial x} = 2\pi x$, and $\Delta x = 0.1\times 10^4$cm (this is the worst case - realistically, we could also assume $\Delta x = 0.05\times 10^4$cm). Taking $\Delta x = 0.1\times 10^4$cm leads to $$\Delta f = 2\pi x \Delta x = 2\pi \times 2.7\times 10^4 \text{cm} \times 0.1\times 10^4\text{cm} = 0.1696 \times 10^9\text{cm}\simeq 0.2 \times 10^9\text{cm}~.$$

Answer 2

A rough and ready way is to say that the radius is $2.7\pm 0.1 \times 10^4\,\rm cm$ which is approximately a $3.7\%$ error.

The area is the radius squared so the error in the area is twice the error in the radius, $2\times 3.7 = 6.4\%$ which then translates into an error of $2.3 \times 10^9 \times 0.064 \approx 0.2 \times 10^9\,\rm cm^2$ in the area.