How do you find the maximum error (Calculus)?

Question

The circumference of a sphere was measured to be 77 cm with a possible error of 0.8 cm. Use differentials to estimate the maximum error in the calculated surface area.

The circumference of a sphere of radius r is C=2pi*r, and its surface area is 4pi*r^2.

Answer 1

C = 2πr, r = C/(2π)
A = 4πr^2 = 4π(C/(2π))^2 = (1/π)C^2
dA/dC = (2/π)C
dA = (2/π)CdC
dA = 2(77)0.8)/π
dA = 39.215 cm^2

Check:
A = (1/π)((77.8^2 - 77^2) = 39.419cm^2

Answer 2

C = 2πr
dC = 2πdr
Thus δC ≈ 2πδr
± 0.8cm ≈ 2πδr
δr = ± 0.8/2πcm
= ± 0.4/πcm

S = 4πr²
dS = 8πrdr
So δS ≈ 8πrδr
= 8π * 77 * ± 0.4/π
= ±1540 cm²