6. (4 points) (a) The edge of a cube was measured to be 6 cm, with a maximum possible error of 0.5 cm. Use a differential to estimate the maximum possible error in computing the volume of the cube. (b) Using a calculator, find the actual error in measuring volume if the radius was really 6.5 cm instead of 6 cm, and find the actual error if the radius was actually 5.5 cm instead of 6 cm. Compare these errors to the answer you got using differentials.

A)   ± 54 cm^3 ( maximum possible error in volume )

B) i) 58.625 cm^3  ii) 49.625 cm^3

Step-by-step explanation:

A) using differential

edge of cube = 6 cm ,  maximum possible error = 0.5 cm

∴ side of cube ( x )= ± 0.5 cm

V = volume of cube

dv /dx = d(x)^3 / dx

∴ dv  = 3x^2 dx ( 1 )

input values into 1

dv = 3(6)^2 * ( ± 0.5 )

    = ± 54 cm^3 ( maximum possible error in volume )

B) Using calculator

actual error in measuring volume when

i) radius = 6.5 cm instead of  6 cm

V1= ( 6.5)^3 = 274.625 ,  V = ( 6)^3 = 216

actual error = 274.625 - 216 = 58.625 cm^3

ii) radius = 5.5cm instead of 6cm

actual error = 49.625 cm^3