Rate of change in the natural and social scienes

Question

The edge of a cube was found to be 30 cm with a possible error in measurement of 0.5 cm. Use differentials to estimate the maximum possible error, relative error, and percentage error in computing the volume of the cube and the surface area of the cube. (Round your answers to four decimal places.)

a.The volume of the cube

maximum possible error

relative error

percentage error

Answer 1

The error in r is .5 cm. The relative error is .5 cm/30 cm = .0167 and the % error is 1.67%

The SA = 6s² for a cube so δSA = 12δs (taking differential of both sides.

Unless adding or subtracting, relative error or percent error is more useful in propagating error:

δSA/SA = 12δs/6s² = 2(δs/s) (Note that generally relative error in z, δz/z, if z = kxⁿ , is n(δx/x))

relative error in SA is 2*(.0167) = .0333 or 3.33%

The absolute error in SA is (SA)*(rel. error in SA) = 6(30 cm)² * .03333 = 180 cm²

Relative error in the volume is 3*(δr/r) = .05 or 5%

and the absolute error is (30cm)³ x (.05) = 1350 cm³

Hope that helped.