Understand the concept of a relation as a mapping between sets, a graph and as a rule or a formula that defines one variable quantity in terms of another.

A veterinary clinic uses a formula to estimate the body mass of a juvenile koala (in kilograms) based on the age of the animal in weeks. The relation is defined by the rule: $$m = 0.8w + 1.2$$ where $m$ is the body mass in kilograms and $w$ is the age in weeks. (a) Calculate the body mass of a koala that is 5 weeks old. [1] (b) Determine the age (in weeks) of a koala whose body mass is 5.2 kg. [1] (c) State the rate of change of body mass with respect to age, and explain what this value represents in the context of the veterinary data. [1]

A relation is defined by the rule $f(x) = \sqrt{x - 4}$ for real numbers $x$. Which of the following correctly describes this relation as a mapping between sets?

A biologist models the population density \(D\) (organisms per square metre) of a particular species in a tidal zone as a function of the distance \(x\) (metres) from the low-tide mark, where \(0 \leq x \leq 50\). The relation is given by \(D(x) = 12 + 8\sin\left(\frac{\pi x}{25}\right)\), \(0 \leq x \leq 50\). (a) Determine the maximum population density and the distance from the low-tide mark at which it occurs. [2 marks] (b) The biologist defines a new variable \(P\) (total population) that represents the cumulative population from the low-tide mark up to a distance \(x\). Determine an expression for \(P(x)\) in terms of \(x\). [2 marks] (c) Determine the total population, to the nearest whole number, across the entire 50-metre zone. [1 mark]