Constructing differential equationsAQA A-Level Maths: Flashcards

Exam questions on Constructing differential equations

1. The number of bacteria $N$ in a culture at time $t$ hours increases at a rate proportional to the number of bacteria present. Let $k$ be the positive constant of proportionality.
   The bacteria are also removed from the culture at a constant rate of $40$ per hour. Write down a differential equation for $N$, using your value of $k$ from part (b).2 marks
2. Water leaks from a tank. The volume of water in the tank is $V$ m$^3$ at time $t$ minutes, and $V$ decreases at a rate proportional to the square root of $V$.
   Water is also pumped into the tank at a constant rate of $0.3$ m$^3$ per minute. Write down the new differential equation and find the volume at which the volume of water stays constant.2 marks
3. A company models the demand $Q$ (in thousands of units) for its product when the price is $\pounds p$. It assumes that $Q$ falls as $p$ rises, and that the rate of change of $Q$ with respect to $p$ is proportional to $Q$.
   Write down a differential equation for $Q$ in terms of $p$ and a positive constant $k$. When $p=5$, $Q=40$ and $\frac{dQ}{dp}=-6$. Find $k$.3 marks