3. as c increases?

1.

2.

3.

Discuss the effects of changing the parameters a, b and c in the following expressions.

4.71. a + b exp(− cx )

2. c + ax exp(− bx )

4.5 Translating into mathematics

Care must be taken when selecting an appropriate mathematical form corresponding to a verbal statement concerning variables. For example, if one variable y is stated to be directly proportional to another variable x (sometimes symbolised as y α x ), the appropriate mathematical expression to represent the relationship is y = kx, where k is the constant of proportionality. The appropriate value of k can be derived if some sample data of x and y values are available.

If y is proportional to x 1 and proportional to x 2 , then the appropriate form could be y = kx 1 x 2 . Note that this implies that y is doubled whenever x 1 or x 2 is doubled for example, which would not be the case for an expression such as y = k 1 x 1 + k 2 x 2 . The second form would be appropriate for a situation such as ‘ y is increased by an amount k 1 for every unit increase in x 1 and by an amount k 2 for every unit increase in x 2 ’. We came across a situation such as this in Example 2.4 in chapter 2.

The statement ‘ y decreases as x increases’ could be interpreted as a linear relationship y = y 0 – ax ( a > 0) or as inverse proportion y = k/x. We would need either to obtain more information about the actual relationship between x and y or to make our own assumptions about which form is more appropriate or whether some other form should be used.

Example 4.5

Suppose than an ice-cream seller at a summer fair guesses that the amount A of ice cream that he will sell is going to be

1.proportional to the number n of people who come to the fair,

2.proportional to the temperature excess over 15°C and

3.inversely proportional to the selling price p.

What would be an appropriate model for A ? Solution

These assumptions would lead to a model of the form A = kn ( T – 15)/ p, where T is the actual temperature in degrees Celsius. Note that the model is valid for T ≥ 15.