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23 October, 01:24

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.

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  1. 23 October, 03:13
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    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
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