A cube is built with inside dimensions of 10 inches. The material is 0.2 inches thick. Use a Taylor series approximation to find the approximate volume of material used.

V (x, y, z) ≈ 61.2 in

Step-by-step explanation:

for the function f

f (X) = x³

then the volume will be

V (x, y, z) = f (X+h) - f (X), where h = 0.2 (thickness)

doing a Taylor series approximation to f (x+h) from f (x)

f (X+h) - f (X) = ∑fⁿ (X) * (X-h) ⁿ/n!

that can be approximated through the first term and second

f (X+h) - f (X) ≈ f' (x) * (-h) + f'' (x) * (-h) ²/2 = 3*x² * (-h) + 6*x * (-h) ²/2

since x=L=10 in (cube)

f (X+h) - f (X) ≈ 3*x² * (-h) + 6*x * (-h) ²/2 = 3*L²*h+6*L*h²/2 = 3*L*h * (h+L)

then

f (X+h) - f (X) ≈ 3*L*h * (h+L) = 3 * 10 in * 0.2 in * (0.2 in + 10 in) = 61.2 in

then

V (x, y, z) ≈ 61.2 in

V real = (10.2 in) ³ - (10 in) ³ = 61 in