Find a functional description of the following data: [latex]\begin{array}{l|llll}x & 1 & 2 & 3 & 4 \\\hline y & 5.1 & 19.5 & 46 & 78\end{array}[/latex]

These data do not lie close to a straight line when plotted on linear or semilog axes. However, they do when plotted on log-log axes. Thus a power function [latex]y=bx^m[/latex] can describe the data. Using the transformations X = log x and Y = log y, we obtain the new data table: [latex]\begin{array}{l|llll}X=\log x & 0 & 0.3010 & 0.4771 & 0.6021 \\\hline Y=\log y & 0.7076 & 1.2900 & 1.6628 & 1.8921\end{array}[/latex] From this table we obtain [latex]\begin{aligned}\sum_{i=1}^4 X_i=1.3803 & \sum_{i=1}^4 Y_i=5.5525 \\\sum_{i=1}^4 X_i Y_i=2.3208 & \sum_{i=1}^4 X_i^2=0.6807\end{aligned}[/latex] Using X, Y, and B = log b instead of x, y, and b in (C.1.1) and (C.1.2), we obtain [latex]\begin{aligned}0.6807 m+1.3803 B &=2.3208 \\1.3803 m+4 B &=5.5525\end{aligned}[/latex] The solution is m = 1.9802 and B = 0.7048. This gives [latex]b = 10^B[/latex] = 5.068. Thus, the desired function is [latex]y = 5.068x^{1.9802}[/latex].

Question C.2.1: Find a functional description of the following data:

System Dynamics

Find a functional description of the following data:

$\begin{array}{l|llll}x & 1 & 2 & 3 & 4 \\\hline y & 5.1 & 19.5 & 46 & 78\end{array}$

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These data do not lie close to a straight line when plotted on linear or semilog axes. However, they do when plotted on log-log axes. Thus a power function $y=bx^m$ can describe the data. Using the transformations X = log x and Y = log y, we obtain the new data table:

$\begin{array}{l|llll}X=\log x & 0 & 0.3010 & 0.4771 & 0.6021 \\\hline Y=\log y & 0.7076 & 1.2900 & 1.6628 & 1.8921\end{array}$

From this table we obtain

$\begin{aligned}\sum_{i=1}^4 X_i=1.3803 & \sum_{i=1}^4 Y_i=5.5525 \\\sum_{i=1}^4 X_i Y_i=2.3208 & \sum_{i=1}^4 X_i^2=0.6807\end{aligned}$

Using X, Y, and B = log b instead of x, y, and b in (C.1.1) and (C.1.2), we obtain

$\begin{aligned}0.6807 m+1.3803 B &=2.3208 \\1.3803 m+4 B &=5.5525\end{aligned}$

The solution is m = 1.9802 and B = 0.7048. This gives $b = 10^B$ = 5.068. Thus, the desired function is $y = 5.068x^{1.9802}$ .