5.14 Differential equations: separation of variablesIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   The mass $G$ grams of algae in a pond at time $t$ days grows at a rate proportional to $G$, so $\frac{dG}{dt}=kG$ where $k>0$ is a constant. Initially $G=50$, and when $t=4$, $G=80$.

   (a)

   Which of the following is the general solution of $\frac{dG}{dt}=kG$?

   [1 mark]

   • A$G=kt+A$
   • B$G=e^{kt}+A$
   • C$G=Ae^{t}$
   • D$G=Ae^{kt}$

   (b)

   Find the value of $k$.

   [1 mark]

   • A$0.150$
   • B$0.118$
   • C$0.470$
   • D$0.0294$

   (c)

   Find the mass of algae after $10$ days, using the unrounded value of $k$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A sample of a radioactive substance has mass $m$ mg at time $t$ years. The mass decreases at a rate proportional to $m$. Initially $m=200$, and after $5$ years $m=150$.

   (a)

   Which differential equation models this situation, where $k>0$?

   [1 mark]

   • A$\frac{dm}{dt}=-km$
   • B$\frac{dm}{dt}=km$
   • C$\frac{dm}{dt}=-kt$
   • D$\frac{dm}{dt}=-\frac{k}{m}$

   (b)

   Separating the variables in $\frac{dm}{dt}=-km$ gives

   [1 mark]

   • A$\int m\,dm=-k\int dt$
   • B$\int\frac{1}{m}\,dm=k\int dt$
   • C$\int\frac{1}{m}\,dm=-k\int dt$
   • D$\int\frac{1}{m}\,dm=-\int kt\,dt$

   (c)

   Find the value of $k$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A boat's engine is switched off at $t=0$. For $t\ge0$ the speed $v$ m s$^{-1}$ of the boat satisfies $\frac{dv}{dt}=-0.05v^2$, with $v=10$ when $t=0$.

   (a)

   Show that $v=\frac{20}{t+2}$.

   [3 marks]

   (b)

   (i) Find the speed of the boat when $t=8$.
   (ii) Find the time at which the speed of the boat is $0.5$ m s$^{-1}$.
   (iii) Explain why, according to this model, the boat never comes to rest.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The temperature $T$ °C of a cup of tea $t$ minutes after it is poured satisfies $\frac{dT}{dt}=-k(T-20)$, where $k>0$ is a constant and $T>20$. Initially $T=90$, and after $5$ minutes $T=60$. A GDC may be used.

   (a)

   (i) By separating variables, show that the general solution is $T=20+Ae^{-kt}$, where $A$ is a constant.
   (ii) Find the value of $A$.
   (iii) Find the value of $k$.

   [6 marks]

   (b)

   Use $T=20+70e^{-kt}$ with the unrounded value of $k=\frac{\ln(7/4)}{5}$.
   (i) Find the temperature of the tea after $12$ minutes.
   (ii) Find the time taken for the tea to cool to $40$ °C.
   (iii) State, with a reason, the temperature the tea approaches in the long term.

   [6 marks]

   Total for question 4: 12 marks

End of questions