Question

What is the percent uncertainty in the area of a circle whose radius is 1.8 × 104 cm?

Answer 1

Here the absolute uncertainty is not given, so assume the last reported digit can vary by 1. That means the absolute uncertainty is 0.1 × 10^4 cm. 

Now, when you perform math operations on the measurements, you have to adjust the absolute uncertainty of the final result. In particular: 
1. If you multiply the measurement by some number "n" (which has zero uncertainty), then you must also multiply the uncertainty by "n". 
2. If you square the measurement, then you must multiply the uncertainty by 2M (where "M" is the amount of the measurement) 
3. And lastly, if you cube the measurement, you must multiply the uncertainty by 3M²So, if the measured radius is "R" (1.8x10^4 cm); and the uncertainty is "Δ" (0.1 × 10^4 cm), then the area is: 

Area = πR² +/- π(2R)Δ 

(Notice I multiplied the Δ by π (according to Rule 1 above) and then by (2R) (according to Rule 2) above.) 

Relative uncertainty = absolute uncertainty / measured value, so: 

Relative uncertainty in area = π(2R)Δ / (πR²) 
= 2Δ/R 
= 0.2 × 10^4 cm / (1.8 times; 10^4 cm) 
(which is about 11%)