The diameter of a spherical ball bearing was measured to be 5 mm with a possible error of 0.09 mm. State answers exactly or to 4 decimal places.

Question

a) Use differentials to estimate the maximum error in the calculated volume.

_____ mm³

b) Use differentials to estimate the relative error in the calculated volume.

_____

Answer 1

First, it's easier to do this with V as a function of d:

V = (4/3)πr³ = (1/6)πd³

Take differential:

δV = (π/2)d² δd plug in d= 5 mm and δd = .09 mm

For δV/V, just divide the RHS by (1/6)πd³ and get δV/V = 3 (δd/d)

Relative error in volume = 3 (.09/5)

Note that in error analysis, you can go straight to this result for power dependence:

if A = cxⁿ δA/A = n(δx/x)

Good luck!