Consider the data of Example C.3.5 . Determine the best-fit value of the coefficient b in the square-root function [latex]f=b h^{1 / 2}[/latex] (1) [latex]\begin{array}{l|lllllllllll}\text{Height} h( cm ) & 11 & 10 & 9 & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 \\\hline \text{Time} t( s ) & 7 & 7.5 & 8 & 8.5 & 9 & 9.5 & 11 & 12 & 14 & 19 & 26\end{array}[/latex]

First obtain the flow rate data in ml/s by dividing the 250 ml volume by the time to fill: [latex]f=\frac{250}{t}[/latex] (2) Referring to Example C.2.3 , whose model is [latex]y = bx^m[/latex], we see here that y = f , h = x, and m = 0.5. From Equation (1) of Example C.2.3 , [latex]b=\frac{\sum_{i=1}^n h_i^{0.5}y_i}{\sum_{i=1}^n h_i}[/latex] (3)

Question C.3.6: Consider the data of Example C.3.5. Determine the best-fit v...

System Dynamics

Consider the data of Example C.3.5. Determine the best-fit value of the coefficient b in the square-root function

$f=b h^{1 / 2}$ (1)

$\begin{array}{l|lllllllllll}\text{Height} h( cm ) & 11 & 10 & 9 & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 \\\hline \text{Time} t( s ) & 7 & 7.5 & 8 & 8.5 & 9 & 9.5 & 11 & 12 & 14 & 19 & 26\end{array}$

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The MATLAB program to carry out these calculations is shown next.

% Enter the data.
h = (1:11);
time = [26, 19, 14, 12, 11, 9.5, 9, 8.5, 8, 7.5, 7];
% Compute flow rate from (2).
flow = 250./time;
% Compute b from (3).
b = sum(sqrt(h).*flow)/sum(h)
% Compute f from (1).
f = b*sqrt(h);
% Compute J, S, and r squared.
J = sum((f - flow).ˆ2)
S = sum((flow - mean(flow)).ˆ2)
r2 = 1 - J/S

The result is a=10.4604 and the flow model is $f=10.4604 \sqrt{h}$ . The quality-of-fit values are J = 5.5495, S = 698.2203, and r² = 0.9921, which indicates a very good fit.