8 The new price of a particular make of car is \(\pounds 10000\). When its age is \(t\) years, the list price is \(\pounds V\). When \(t = 5 , V = 5000\).
Aloke, Ben and Charlie all run outlets for used cars. Each of them has a different model for the depreciation.
- Aloke claims that the rate of depreciation is constant. Write this claim as a differential equation.
Solve the differential equation and hence find the value of a car that is 7 years old according to this model.
Explain why this model breaks down for large \(t\). - Ben believes that the rate of depreciation is inversely proportional to the square root of the age of the car. Express this claim as a differential equation and hence find the value of a car that is 7 years old according to this model.
Does this model ever break down? - Charlie believes that a better model is given by the differential equation
$$\frac { \mathrm { d } V } { \mathrm {~d} t } = k V$$
Solve this differential equation and find the value of the car after 7 years according to this model.
Does this model ever break down? - Further investigation reveals that the average value of this particular type of car when 8 years old is \(\pounds 3000\).
Find the value of \(V\) when \(t = 8\) for the three models above. Which of the three models best predicts the value of \(V\) at this time?