Question

The volume V of a cylinder is computed using the values 8.8m for the diameter and 5.8m for the height. Use the linear approximation to estimate the maximum error in V if each of these values has a possible error of at most 8%.

Answer 1

The maximum error is approximately Ev=24%

Explanation:

the volume of the cylinder V is

V= π/4*H*D²

where H= height and D= diameter

the variation of V will be

dV = (∂V/∂H)*dH + (∂V/∂D)*dD

dV = π/4*D²*dH +π/2*H*D*dD

if we divide by the volume V

dV /V = (π/4*D²*dH +π/2*H*D*dD )/( π/4*H*D²) = dH/H + 2*dD/D

dV /V = dH/H + 2*dD/D

then we can approximate

error in V= Ev= ΔV/V ≈ dV/V

error in H= Eh=ΔH/H ≈ dH/H

error in D= Ed=ΔD/D ≈ dD/D

thus

Ev= Eh + 2*Ed

since Ed=Eh=E=8%

Ev= Eh + 2*Ed =3*E=3*8%=24%

Ev= 24%

therefore the maximum error is approximately Ev=24%