Differential Equations · Paper 1 — Free Maths AI SL Practice Questions

The population of a small island, $P$ (in thousands), is modelled by the differential equation $$\frac{dP}{dt} = 0.02P$$ where $t$ is the time in years after 1 January 2000. The general solution of $\frac{dP}{dt} = kP$ is $P = Ae^{kt}$, where $A$ and $k$ are constants.

A cup of coffee cools according to Newton's Law of Cooling. The temperature $T$ (in °C) of the coffee at time $t$ minutes satisfies the differential equation $$\frac{dT}{dt} = -0.1(T - 22)$$ where $22$°C is the room temperature.

A population of bacteria in a laboratory culture grows over time. The rate of change of the population, $P$ (in thousands), at time $t$ (in hours) is modelled by the differential equation $$\frac{dP}{dt} = kP(10 - P)$$ where $k$ is a positive constant.

A chemical processing plant uses a holding tank that initially contains $5000$ litres of a chemical solution. The solution contains $200\,\text{kg}$ of dissolved salt. A pipe feeds a new mixture into the tank at a rate of $40$ litres per minute; this incoming mixture contains $1.5\,\text{kg}$ of salt per litre. The well-stirred mixture leaves the tank at the same rate of $40$ litres per minute. Let $S(t)$ represent the amount of salt (in kg) in the tank after $t$ minutes.