does anyone know this problem??

Question

The radius of a circular disk is given as 28 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?

What is the percentage error?

Answer 1

The measurment error is ∆R, and ∆A = ∂A/∂R * ∆R

since A = πR^2, ∂A/∂R = 2πR; the error in area is then

∆A = 2πR∆R ∆R = 0.2cm R = 28cm ∆A = 35.19 cm^2

The relative error is ∆A/A, = 2*∆R/R; the relative error in area is twice the relative error in radius. ∆R/R = 0.00714 so ∆A/A = 0.0143

Percent error is relative error times 100 or 1.43%

Answer 2

let A = area, r = radius of disk
we know A = pi*r^2,
setting in differential form we have,
dA = (2*pi*r) dr
note that in this case, dA and dr represent small (finite) increments in A and r respectively.

a) max error of A is expressed as dA, which is
=> dA = (2*pi*28) (0.2) = 11.2*pi cm^2 = 35.2 cm^2

b) relative error = [max. A - actual A]/actual A
= 35.2/(pi*(28^2))
=0.0143

Percentage error = [(max. A - actual A)/actual A] * 100
= relative error*100%
= (35.2)/(pi*(28^2))*100
=1.43 %