Question

The measurement of the radius of a circle is 16 inches, with a possible error of .25 inches.

Use differentials to approximate the possible propagate error in computing the area of the circle.

Answer 1

dr = .25 - the error in the radius

a = pi r^2
da = 2pi r dr
so, as you found, the variation in area is 8pi.

the relative error is thus

da/a = 8pi/(256pi) = 1/32 = 3.125%

Answer 2

a = pi r^2
da = 2pi r dr

now just plug in your numbers.