The radius of a circular disk is given as 22 cm with a maximum error in measurement of 0.2cm.
(a) use differentials to estimate the maximumerror in the calculated area of the disk. (round the answer to twodecimal places.)
cm2

(b) what is the relative error? (round the answer to four decimalplaces.)

what is the percentage error? (round the answer to two decimalplaces.)
%

The maximum error in the calculated in the area is about

The relative error is 0.02

The percentage error is 1.82%

Step-by-step explanation:

(a) Differentials are infinitely small quantities. Given a function we call and differentials and the relationship between them is given by,

Let be the radius of the disk and its area . If the error in the measured valued of is denoted by , then the corresponding error in the calculated value of A is , which can be approximated by the differential

We know that and , substituting into the above differential we get

The maximum error in the calculated in the area is about

(b) The definition of relative error is

To find the relative error you need to divide the error by the total area

(c) To find the percentage error you need to apply this formula

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4