Constructing differential equationsAQA A-Level Maths: Subtopic test

1. 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.

   (a)

   Which differential equation models the situation?

   [1 mark]

   • A$\frac{dN}{dt}=\frac{k}{N}$
   • B$\frac{dN}{dt}=kt$
   • C$N=kt$
   • D$\frac{dN}{dt}=kN$

   (b)

   At one moment there are $2000$ bacteria and the number is increasing at $300$ bacteria per hour. Find $k$.

   [1 mark]

   • A$0.15$
   • B$6.67$
   • C$1.5$
   • D$0.015$

   (c)

   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]

   Total for question 1: 4 marks

2. 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 $V$.

   (a)

   Which differential equation models the situation, where $k$ is a positive constant?

   [1 mark]

   • A$\frac{dV}{dt}=k\sqrt V$
   • B$\frac{dV}{dt}=-k\sqrt V$
   • C$\frac{dV}{dt}=-kV^2$
   • D$\frac{dV}{dt}=-\frac{k}{\sqrt V}$

   (b)

   When $V=16$ the volume is decreasing at $0.8$ m$^3$ per minute. Find $k$.

   [1 mark]

   • A$0.05$
   • B$0.8$
   • C$0.2$
   • D$12.8$

   (c)

   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]

   Total for question 2: 4 marks

3. 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 $Q$.

   (a)

   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]

   (b)

   The revenue is $R=pQ$ thousand pounds. Show that $\frac{dR}{dp}=Q(1-kp)$, and find the price at which $\frac{dR}{dp}=0$, using $k=0.15$. Give your answer to the nearest penny.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A parachutist of mass $m$ kg falls vertically from rest. Her speed after $t$ seconds is $v$ m s$^{-1}$. Take downwards as positive and $g=9.8$ m s$^{-2}$.

   (a)

   The air resistance on the parachutist is modelled as $\lambda v$ newtons, where $\lambda$ is a positive constant. (i) Show that $\frac{dv}{dt}=g-\frac{\lambda}{m}v$. (ii) Find the terminal speed in terms of $m$, $g$ and $\lambda$. (iii) Find the terminal speed when $m=60$ and $\lambda=12$.

   [6 marks]

   (b)

   A second model takes the air resistance as $\mu v^2$ newtons, where $\mu$ is a positive constant. (i) Construct a differential equation for $v$. (ii) Find the terminal speed when $m=60$ and $\mu=0.3$, correct to 3 significant figures.

   [6 marks]

   Total for question 4: 12 marks

End of questions