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

Answer 1

There are two ways of answering this question

Calculus Let A denote the area and r the radius.

Then A = pi*r2 = pi*1.8*104cm = 1,017,876*103 cm2

Now "error" = dA = pi*2r*dr where dr is the error in the measurement of the radius = 0.05*104 = 500 cm.

So dA = 56,549*103 cm2.

Therefore, percentage error = 100*dA/A = 5.55... (recurring) %

Explicit calculation

r = 1.8*104 cm. Therefore the range for radius is 17,500 to 18,500 cm.

That gives an area of 1,017,876*103 cm2 with a range of

962,113*103 to 1,075,210*103 cm2

That gives an average absolute error of 56,549*103 cm2 and as before, the percentage error is 5.55... %.