Constructing differential equationsEdexcel A-Level Maths: Subtopic test

1. 1
   A population of insects has $N$ individuals at time $t$ days. The rate of increase of $N$ is proportional to $N$. Initially $N=500$ and the population is increasing at $60$ insects per day.

   (a)

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

   [1 mark]

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

   (b)

   Find the value of $k$ in the differential equation.

   [1 mark]

   • A$8.3$
   • B$0.012$
   • C$0.12$
   • D$30\,000$

   (c)

   Find the rate of increase of the population when $N=2000$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A spherical ball of ice melts so that its radius $r$ cm decreases at a rate that is inversely proportional to the square of the radius. Time $t$ is measured in minutes. When $r=2$, the radius is decreasing at $0.1$ cm per minute.

   (a)

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

   [1 mark]

   • A$\frac{dr}{dt}=\frac{k}{r^{2}}$
   • B$\frac{dr}{dt}=-\frac{k}{r^{2}}$
   • C$\frac{dr}{dt}=-kr^{2}$
   • D$\frac{dr}{dt}=-\frac{k}{r}$

   (b)

   Find the value of $k$ in the differential equation.

   [1 mark]

   • A$0.025$
   • B$0.2$
   • C$0.05$
   • D$0.4$

   (c)

   Find the rate at which the radius is decreasing when $r=0.5$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Water flows into a tank at a constant rate of $0.6$ m$^3$ per minute and leaves through a hole in the base at a rate of $0.1\sqrt h$ m$^3$ per minute, where $h$ m is the depth of the water at time $t$ minutes. The tank has a horizontal cross-section of area $2$ m$^2$, so the volume of water in the tank is $V=2h$ m$^3$.

   (a)

   Show that $\frac{dh}{dt}=0.3-0.05\sqrt h$.

   [3 marks]

   (b)

   Find the depth at which the depth of water is not changing, and the rate at which the depth is changing when $h=16$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A patient is given a drug by a drip that delivers $5$ mg of the drug per hour. The body removes the drug at a rate proportional to the amount $x$ mg of the drug in the body at time $t$ hours. When $x=40$ the amount of the drug in the body is not changing.

   (a)

   (i) Show that $\frac{dx}{dt}=5-kx$, where $k$ is a positive constant.
   (ii) Find the value of $k$.
   (iii) Find the rate at which $x$ is changing when $x=16$.

   [6 marks]

   (b)

   A second model has the body removing the drug at a rate proportional to $\sqrt x$, with the same drip rate and the same condition that $x=40$ is not changing. Construct the differential equation for this model, and compare the rates of change of $x$ predicted by the two models when $x=16$.

   [6 marks]

   Total for question 4: 12 marks

End of questions