Linear Approximation and Differentials

Question

The side length of a cube is measured to be x = 1.3 feet, with an error of at most 1 inch.

(A) (No calculus) Determine the difference, in ft^3, between the volume of the cube

computed with the measured side length and the volume computed with the largest

(resp., smallest) value for the side length in the error range. Remember first to

convert all lengths to feet! Round your final answers to three digits after the

decimal point.

(B) Use differentials to estimate the error in calculating the volume of the cube using

the measured value and error estimate for the side length of the cube. Round your

final answer to three digits after the decimal point.

Answer 1

Let the side length be represented as "x". Then:

V = x³

If x = 1.3, then V = 1.3³ = 2.197 ft³

If the side length reached its max at 1.3 ft + 1 inch, then the side length would be 1.3 + 1/12 = 1.383333 and then:

V = 1.383333³ = 2.647 ft³

So the difference is 2.647 - 2.197 = 0.450 ft³

Now, using differentials:

V = x³

dV/dx = 3x²

ΔV/Δx ≈ 3x²

ΔV ≈ (3x²)Δx

ΔV ≈ [3(1.3)²](1/12) = 0.360 ft³