Question

The radius of a circular disk is given as 21 cm with a maximum error in measurement of 0.2 cm.

A) Use differentials to estimate the maximum error in the calculated area of the disk.
B) What is the relative error?
C) What is the percentage error?

Answer 1

a)
`\Delta A \approx 26.389\,cm^(2)`, b)
`r_(A) \approx 0.019`, c)
`\delta = 1.9\,\%`

Explanation:

a) The area of the circular disk is modelled after this expression:

`A = \pi \cdot r^(2)`

The total differential is given by the following formula:

`\Delta A = 2\pi r \cdot \Delta r`

The maximum absolute error in the calculated area of the disk is:

`\Delta A = 2\pi \cdot (21\,cm)\cdot (0.2\,cm)`

`\Delta A \approx 26.389\,cm^(2)`

b) The relative error is given by:

`r_(A) = (\Delta A)/(A)`

`r_(A) = (26.389\,cm^(2))/(\pi \cdot (21\,cm)^(2))`

`r_(A) \approx 0.019`

c) The percentage error is:

`\delta = r_(A)* 100\,\%`

`\delta = 0.019 * 100\,\%`

`\delta = 1.9\,\%`