propogated error in cm^2 and relative error

Question

The radius of a circular disk is measured as  30cm with a maximum error in measurement of +/- 4 cm. Use differentials to estimate the propagated error and the relative error in the calculated area of the disk.

Answer 1

Area = A = pi * r²

^{Take differentials of both sides. On the right side this means taking the derivative wrt r and then putting dr on the end.}

^{dA = 2pi * r * dr}

Now use the differentials to approximate delta A and delta r

delta A ≅ 2pi * r * delta r

delta A ≅ 2pi. * 30cm * (+/- 4cm)

delta A ≅ +/- 753.98 sq.cm. (approximate calculator answer) This is the propagated error.

(Since 4cm. is only 1 significant digit this should be rounded to +/- 800 sq.cm. but most calculus books don't do this.)

To get the relative error divide the propagated error by measured area.

A = pi * r² = pi * 30cm² = 2827.43sq.cm.

Relative error ≅ dA/A = +/- 753.98sq.cm/2827.43sq.cm = +/- .267

Or expressed as a percent: Relative error ≅ +/- 26.7%