Question C.1.2: A hole 6 mm in diameter was made in a translucent milk conta...
A hole 6 mm in diameter was made in a translucent milk container (Figure C.1.7). A series of marks 1 cm apart was made above the hole. While adjusting the tap flow to keep the water height constant, the time for the outflow to fill a 250-ml cup was measured (1 ml=10^{−6} m³). This was repeated for several heights. The data are given in the following table.
\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}
Obtain a functional description of the volume outflow rate f as a function of water height h above the hole.
First obtain the flow rate data in ml/s by dividing the 250 ml volume by the time to fill:
f=\frac{250}{t}
A plot of the resulting flow rate data is shown in Figure C.1.8. There is some curvature in the plot, so we rule out the linear function. Common sense tells us that the outflow rate will be zero when the height is zero, so we can rule out the exponential function because it cannot pass through the origin.
The log-log plot shown in Figure C.1.9 shows that the data lie close to a straight line, so we can use the power function to describe the flow rate as a function of height. Thus, we can write
f=b h^m
The straight line shown can be drawn by aligning a straightedge so that it passes near most of the data points (note that this line is subjective; another person might draw a different line). Next we select two points on the straight line to find the values of b and m. The two points indicated by an asterisk were selected to minimize interpolation error because they lie near grid lines. The accuracy of the values read from the plot obviously depends on the size of the plot. The values of the points as read from the plot are (1, 9.4) and (8, 30). Using the equations developed previously to compute b and m (with h replacing x and f replacing y), we have
\begin{aligned}m &=\frac{\log (30 / 9.4)}{\log (8 / 1)}=0.558 \\b &=9.4\left(1^{-0.558}\right)=9.4\end{aligned}
Thus, the estimated function is f=9.4 h^{0.558}, where f is the outflow rate in ml/s and the water height h is in centimeters. The plot of f versus h is shown in Figure C.1.10. From this we can see that the function provides a reasonably good description of the data. In Section C.2 we will discuss how to quantify the quality of this description.
