Answer to Question #153480 in Quantitative Methods for usman

Question #153480

Given the table of values

x: 50 52 54 56

3√x : 3.684 3.732 3.779 3.825

Use Lagrange’s formula to find x when 3√x = 3.756,


1
Expert's answer
2021-01-07T15:32:36-0500

"\\def\\arraystretch{1.5}\n \\begin{array}{c:c:c:c:c}\n x: & 50 & 52 & 54 & 56 \\\\ \\hline\n y=\\sqrt[3]{x}: & 3.684 & 3.732 & 3.779 & 3.825\\\\\n \\hdashline\n \n\\end{array}"

By Lagrange’s interpolation formula for inverse interpolation, we have:

"x=f^{-1}(y)= \\frac{(y-y_1)(y-y_2)(y-y_3)}{(y_0-y_1)(y_0-y_2)(y_0-y_3)}x_0+ \\frac{(y-y_0)(y-y_2)(y-y_3)}{(y_1-y_0)(y_1-y_2)(y_1-y_3)} x_1+ \\frac{(y-y_0)(y-y_1)(y-y_3)}{(y_2-y_0)(y_2-y_1)(y_2-y_3)}x_2+ \\frac{(y-y_0)(y-y_1)(y-y_2)}{(y_3-y_0)(y_3-y_1)(y_3-y_2)} x_3"

We put "y=3.756" :

"x(3.756)=f^{-1}(3.756)= \\frac{(3.756-3.732)(3.756-3.779)(3.756-3.825)}{(3.684-3.732)(3.684-3.779)(3.684-3.825)}50+ \\frac{(3.756-3.684)(3.756-3.779)(3.756-3.825)}{(3.732-3.684)(3.732-3.779)(3.732-3.825)} 52+ \\frac{(3.756-3.684)(3.756-3.732)(3.756-3.825)}{(3.779-3.684)(3.779-3.732)(3.779-3.825)}54+ \\frac{(3.756-3.684)(3.756-3.732)(3.756-3.779)}{(3.825-3.684)(3.825-3.732)(3.825-3.779)} 56=-\\frac{529}{8930}\\cdot 50+\\frac{1587}{2914}\\cdot 52\n+\\frac{2592}{4465}\\cdot 54-\\frac{96}{1457}\\cdot 56\\approx 53.016"


Answer: "x\\approx 53.016" .


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