- The mass, \(A\) kg, of algae in a small pond, is modelled by the equation
$$A = p q ^ { t }$$
where \(p\) and \(q\) are constants and \(t\) is the number of weeks after the mass of algae was first recorded.
Data recorded indicates that there is a linear relationship between \(t\) and \(\log _ { 10 } A\) given by the equation
$$\log _ { 10 } A = 0.03 t + 0.5$$
- Use this relationship to find a complete equation for the model in the form
$$A = p q ^ { t }$$
giving the value of \(p\) and the value of \(q\) each to 4 significant figures.
- With reference to the model, interpret
- the value of the constant \(p\),
- the value of the constant \(q\).
- Find, according to the model,
- the mass of algae in the pond when \(t = 8\), giving your answer to the nearest 0.5 kg ,
- the number of weeks it takes for the mass of algae in the pond to reach 4 kg .
- State one reason why this may not be a realistic model in the long term.