- The value of a car, \(\pounds V\), can be modelled by the equation
$$V = 15700 \mathrm { e } ^ { - 0.25 t } + 2300 \quad t \in \mathbb { R } , t \geqslant 0$$
where the age of the car is \(t\) years.
Using the model,
- find the initial value of the car.
Given the model predicts that the value of the car is decreasing at a rate of \(\pounds 500\) per year at the instant when \(t = T\),
- show that
$$3925 \mathrm { e } ^ { - 0.25 T } = 500$$
- Hence find the age of the car at this instant, giving your answer in years and months to the nearest month.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
The model predicts that the value of the car approaches, but does not fall below, \(\pounds A\).
- State the value of \(A\).
- State a limitation of this model.