6 The value \(\pounds V\) of a car \(t\) years after it is new is modelled by the equation \(V = A \mathrm { e } ^ { - k t }\), where \(A\) and \(k\) are positive constants which depend on the make and model of the car.
- Brian buys a new sports car. Its value is modelled by the equation
$$V = 20000 \mathrm { e } ^ { - 0.2 t } .$$
Calculate how much value, to the nearest \(\pounds 100\), this car has lost after 1 year.
- At the same time as Brian buys his car, Kate buys a new hatchback for \(\pounds 15000\). Her car loses \(\pounds 2000\) of its value in the first year. Show that, for Kate's car, \(k = 0.143\) correct to 3 significant figures.
- Find how long it is before Brian's and Kate's cars have the same value.