Step by step
The question
Question: The circle has radius 4 cm, measured with a possible error of 0.1 cm.
Estimate the percentage error in the area.
1 Draw and name
The radius is r = 4 cm.
The possible error in it is dr = 0.1 cm: a small change.
2 Find dA
The area of a circle is A = π r².
Take the derivative, times dr: dA = 2πr dr.
The change in the area is about the derivative times the change in the radius.
3 Put the numbers in
Put in r = 4 and dr = 0.1.
That is 0.8π, about 2.51 cm².
Careful!
A small error in the radius makes a bigger error in the area.
4 The error
The percentage error compares the error with the whole area.
The whole area is about 50.27 cm².
So the percentage error is about 5%.
5 Check
Let's check with the real change: the area at 4.1, minus the area at 4, is 2.54 cm².
Close to our 2.51.
Right!
Let's look back.
First: draw and name.
Then: find dA.
Then: put the numbers in.
Then: the error.
Last: check.