Exercises

A variable w is related to two other variables x and y in such a way that w is proportional to x and also proportional to y. Which of the following correctly expresses the relationship?

4.81. w = a ( x + y ), a constant.

2.w = ax + by, a and b constants.

3.w = axy, a constant.

A variable y depends on two other variables w and z. The following facts are known.

1.When w increases, y decreases.

2.When z increases, y also increases.

3.When w and z are both zero, y is also zero.

Which of the following models are consistent with facts (i), (ii) and (iii)?

4.9

1. y = aw + bz, a and b constants > 0.

2. y = bz − aw + c, a, b and c constants > 0.

3.

4.y = cwz, c constant > 0.

5.y = az − bw, a and b constants > 0.

When a fluid flows through a pipe, the frictional force F between the pipe wall and the fluid is assumed to be proportional to the length L of the pipe and the square of the fluid speed U. It is also

4.10assumed to be inversely proportional to the diameter D of the pipe. Write down an expression for F in terms of L, U and D and involving a constant k. What are the dimensions of k ? In what units would k be measured?

4.6 Choosing mathematical functions

Consider the following common situations from everyday life.

Case 1 It rains suddenly for 20 min, the rain increasing in intensity before stopping.

Case 2 You are selling ice cream on a hot day; demand is at its greatest when the temperature is highest.

Case 3 You need helpers for the youth club jumble sale on Saturday morning; sales pressure is at its greatest immediately you open for business.

In all three cases, we can see that the effects represented are probably not constant throughout the events. If these phenomena occur as constituent parts of larger mathematical models, then we shall

have to speculate on the behaviour of each before inserting the mathematical representation into our model. For example, the selling of ice cream could be investigated from the profit and loss point of view over a long period. Rainfall rates will contribute to models on reservoir collection, drainage problems and so on.

The common feature of the three situations above is that within each there is a rate of change in some quantity which is not constant over a period of time. Also in each case, the accumulated totals of the quantities are probably known, i.e. in Case 1 we can measure the total amount of rain collected in 20 min, in Case 2 we have a rough idea of how much ice cream can be sold per day, and in Case 3 we can count, or estimate from past experience, how many people will come to the jumble sale. Therefore, if the ‘quantity’ is denoted by Q, and time is denoted by t, then we have, from calculus, the relation

From the modelling point of view, it is the form d Q /d t = Q ′( t ) that is of most interest. In section 4.4, we have shown how the behaviour of certain mathematical functions can be represented. Now we shall build on that work by looking at various possibilities for representing the three situations outlined above.

Case 1

Suppose that, after 20 min of heavy rain, in of rain has fallen. You immediately think of rain falling over an area, but weather centres prefer to use the height measurement only. Our objective is to model the rate of rainfall by a suitable function R ′( t ), and there are of course many possible choices.

1.Steady continuous rain at a constant rate. This is shown in Figure 4.11.

2.It could start to rain slowly before picking up to a maximum and then subsiding again, over the 20 min period, with the total amount collected remaining at in. If a steady linear increase in the rate is taken followed by a similar decrease, then we arrive at Figure 4.12. Now the maximum rainfall rate is 0.05 in min −1 so that we still get a total of in collected (check the area under the graph).

Figure 4.11

There is no need to stop at the situation modelled by (b), particularly if the rain is more or less steady apart from the start and finish. Suppose that the rate increases steadily for 2 min and then remains constant for the next 16 min before decreasing again until it stops after 20 min. This is represented in Figure 4.13. The maximum rainfall rate would then be approximately 0.028 in min−1. (How did we get that figure?)

Now, as a constituent part of some larger model, perhaps resulting in a differential equation, it is the functional form of R ′( t ) that is needed.

For (a), we have

For (b) we have the functional form given by two separate linear forms, each holding over different ranges of time t:

Note that the two formulae agree at t = 10 (otherwise we would have a discontinuous function).

For (c) we have the functional form given by three separate parts, each holding over different ranges of t values:

(Note that we may not need all the decimal places.)

Figure 4.12

Figure 4.13

Again, check that these forms match up at the change-over points t = 2 and t = 18; so there are no breaks. It is quite common in modelling to find that a function is best represented by a formula made up from different forms in different parts of the range as in this example. Do not think that one single formula covering the whole range has to be found (although this is very convenient if it can be done).

Case 2

For the ice-cream sales, much the same set of mathematical functions could be used. However, it is more likely here that sales will build up to a peak in the middle of the day and then subside again, in a ‘smoother’ way. Note that we are modelling the amount of ice cream sold as a continuous variable although it is sold in discrete lumps and is therefore a discrete variable. We assume that the approximation is sufficiently accurate for the purpose of the model. The amount of rain collected at any time is of course very accurately represented as a continuous variable.

Suppose data are given that 1000 ice-cream cones are sold on a hot day. The kiosk is open for 8 h continuously from 10.00 am until 6.00 pm. Let I(t) represent the number of ice-cream cones sold up to time t measured in hours with t = 10 corresponding to 10.00 am. To model the gradual rise in sales, we are interested first in selecting a suitable function for the rate I ′( t ) of sales. One possible selection is based on the use of the sine function.

From our knowledge of the behaviour of the sine function given in section 4.4, we select I ′( t ) = a sin (ω t ) which has a period 2π/ω and amplitude a. A moment's thought tells us that this is inadequate since sin(ω t ) will take up negative values for certain t values. A better choice will be provided by I ′( t ) = a sin 2 (ω t ), but t has to be scaled to run from 10:00 to 18:00 h and we want I ′ and I to be zero at each end value of t. The required form of behaviour is shown in Figure 4.14.

Note that we do not choose a quadratic curve (shown as a broken curve in the figure) because it has steep slopes at the end points. After some thought, we decide on