Question

The edge of a cube was found to be 30 cm with a possible error in measurement of 0.4 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.)

Answer 1

A) Maximum error = 144 cm²

B) Relative error = 0.0267

C) Percentage error = 2.67%

Explanation:

Let x be one side of the cube. Thus, area of one face = x²

Thus, since there are 6 faces, Total area A(x) = 6x²

Now, let's differentiate.

dA/dx = 12x

dA = 12x. dx

When dx is very small like in the question,

ΔA ≈ 12x Δx

From the question, x = 30cm and Δx = 0.4

ΔA ≈ 12 x 30cm x 0.4cm = 144 cm²

Maximum error = 144 cm²

Relative error is given as;

Relative error = Error/surface area

And Surface area = 6x² = 6 x 30² = 5400 cm²

Thus; relative error = 144/5400 = 0.0267

Percentage error is given as;

% error = relative error x 100 = 0.0267 x 100 = 2.67%