Question

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.

Answer 1

V(x,y,z) ≈ 61.2 in

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