Question

The measurement of the edge of a cube is measured to be 12 inches, with a possible error of 0.03 inch. Use differentials to approximate the possible error and the relative error in computing the following values. (Round your answers to three decimal places.)
(a) the volume of the cube
possible errort X __________
relative error + ________
(b) the surface area of the cube
possible error__________.
relative error + X in_________

Answer 1

Answer:

A) possible error = 12.992

Possible values for the volume [1715,007,1740.992]

B) possible error = 4.325

Possible values for the surface area [859,675,868,325]

Step-by-step explanation:

The volumen of a cube is the cube of the length of a side. Thus, the meassurement of the volume is 12³ = 1728 inches³. To calculate the error, lets call Y the real value of the length,

Y = X + e, where e is the error and X = 12.

Y³ = (X+e)³ = X³ + 3X²e + 3Xe² + e³

the error is 0.03 inches, so we have that

Y³ = X³ + 12.96+0.0324+ 0.000027 = X³+12.992427 = X³+12.992 (rounded to 3 decimal places).

Thus, the absolute possible error is Y³- X³ = 12.992, and the volume of the cube is contained in the interval [X³-12.992,X³+12.992] = [1715,007,1740.992]

B) The surface area is 6 X². With X meassured as 12, the surface area is meassured by 6*12² = 864.

To calculate the absolute error, we take Y = X+e, with X and e as before. We have

6 Y² = 6*(X+e)² = 6X²+12Xe+6e² = 6X²+4.32+0.0054 = 6X²+4.3254

Hence, the absolute error is bounded by 6Y²-6X² = 4.3254 rounded by 4.325, and the value of the surface area is contained in the interval [6X² -4.325, 6X²+4.325] = [859,675, 868,325]