Use differential to estimate the maximum error in the calculated a) surface area ; b) volume?

Question

The circumference of a sphere was measured to be 84 cm with a possible error of .5mm.

Answer 1

C = Circumference, S = Surface Area, V = Volume, R = Radius, pi = 3.14159..

C = 2*pi*R = 84, So, R = C/(2*pi) = 84/(2*pi) = 13.369 cm

dC = 2*pi*dR So, dR = dC/(2*pi) = (.05)/(2*pi) = .00796 cm

S = 4*pi*R² = 2246 cm², V = (4/3)*pi*R³ = 10,008.9 cm³

dS = 8*pi*R*dR = 8*pi*(13.369)*(.00796) = 2.67 cm² (error)

dV = 4*pi*R²*dR = 4*pi*(13.369)²*(.00796) = 17.88 cm³ (error)