Maths Modelling
- 6. Example 1: The linear function (or graph of y = mx + c) A common way of introducing the linear function is to use the graph of a straight line. One would normally state that the graph of the linear function y = mx + c is a straight line with gradient m and y-intercept c. This context-free way of teaching is efficient and neat. However, it may be more interesting to see how such a graph and function can actually arise from a real practical situation.
- 7. Consider the following situation where water flows from a tap into a measuring cylinder at a constant rate (as depicted in Figure 2). Suppose we wish to construct a model to show how the water level changes with time so that we can predict how long it would take to fill the whole cylinder. The water level at various points in time can be read off the measuring cylinder. The data is recorded in the form of a graph as shown.
- 9. From the data, we can now try and guess the relationship between the water level,y, and the time after the tap is turned on, t, assuming that the initial water level is c. It is not hard to see that the water level, y, at any time t should be c plus some positive number, and this positive number should depend on t. Eventually, the model obtained should look something like y = c + kt.
- 10. By modelling this simple physical situation, the linear relationship could “come alive”. The linear function is given some context and the graph actually represents something real and physical. Furthermore, the process of modelling would hopefully enable the learner to appreciate other related concepts. For instance, we get a steeper gradient of the graph when the rate of water flowing from the tap is increased.
- 11. Example 2: Biggest box problem Suppose we intend to make an open-top box using a square piece of card of side s by cutting a square (of side, say x) from each corner of the card (see Figure 3). The resulting piece is then folded to form the box.
- 13. The question is: what should x be if we wish to make the biggest box (in terms of volume)? There are several approaches to this problem. Here, two are described.
- 14. The empirical model involves actually constructing the boxes and taking measurements. This has to be done systematically just like in performing a scientific experiment. Since we are particularly interested in the relationship between the size of the smaller square (i.e. x) and the volume of the box, we systematically make boxes using different values of x.
- 15. The sides of the box can then be measured and volume calculated for each case. Alternatively, the volume may be estimated by first pouring sand to completely fill the box. The amount of sand used can be measured using a measuring cylinder. Still another variant could be to weigh the sand instead. Whichever approach is used, the results can be presented in the form of a graph (Figure 4): Figure 4: Graph of Volume against x (from data) A “curve of best fit” is then sketched to locate and estimate x that gives the maximum volume.
- 17. An analytical or theoretical model may also be constructed to solve the problem. This approach is more abstract and involves the use of algebra and geometry. We model the box by a geometric diagram (such as the one in Figure 5). We then find the volume of the box in terms of the dimensions s and x. It is not hard to see that the volume of the box, V is given by or . Suppose the original square cards have sides of dimension, say, s = 10 cm. Then, we have ..
- 18. This is perhaps a good point to introduce the cubic function. In this particular case, the function models the relationship between the volume of the box and the size of the cut-off square. It now remains for us to find the value of x that makes V maximum. How this is done depends on the mathematical ability and maturity of the learner. For instance, a student familiar with calculus may choose to find the derivative and the turning point of the function to obtain the maximum. Another may use a graphing tool to plot a graph of V against x to estimate the maximum. Figure 5 shows a plot generated from the popular graphing tool.