2.9 Further modelling functionsIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   A radioactive isotope decays so that its mass $m$ mg after $t$ hours is modelled by $m=80\left(\frac12\right)^{t/6}$, for $t\ge0$.

   (a)

   Write down the half-life of the isotope.

   [1 mark]

   • A3 hours
   • B6 hours
   • C12 hours
   • D80 hours

   (b)

   Find the mass of the isotope after 18 hours.

   [1 mark]

   • A$10$ mg
   • B$26.7$ mg
   • C$20$ mg
   • D$4.44$ mg

   (c)

   Find the time at which the mass of the isotope is 5 mg.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The height $H$ metres of a tree $x$ years after planting ($x\ge1$) is modelled by $H=a+b\ln x$, where $a$ and $b$ are constants. When $x=1$, $H=2$, and when $x=e^{2}$, $H=5$.

   (a)

   Find the value of $a$.

   [1 mark]

   • A$5$
   • B$3$
   • C$1.5$
   • D$2$

   (b)

   Find the value of $b$.

   [1 mark]

   • A$3$
   • B$0.75$
   • C$1.5$
   • D$2.5$

   (c)

   Use your GDC to find the age at which the tree is 7 m high.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The depth $d$ metres of water at a harbour entrance, $t$ hours after midnight, is modelled by $d(t)=2.5\sin\left(\frac{\pi}{6}(t-1)\right)+6$, for $0\le t\le24$, where the angle is in radians.

   (a)

   Write down (i) the minimum depth, (ii) the period of the model, (iii) the phase shift.

   [3 marks]

   (b)

   A ship needs a depth of at least 7 m. Use your GDC to find the length of time in the first 12 hours of the day for which $d(t)\ge7$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The population $N$ of rabbits on a small island, $t$ years after 1 January 2020, is modelled by $N(t)=\dfrac{480}{1+11e^{-0.9t}}$, for $t\ge0$.

   (a)

   (i) Find $N(0)$ and state what it represents.
   (ii) Write down the carrying capacity of the island and explain what it means.
   (iii) Use your GDC to find the time at which the population reaches 300.

   [6 marks]

   (b)

   A second model for the first few years is the piecewise function
   $M(t)=40+30t$ for $0\le t<2$, and $M(t)=480-\frac{a}{t}$ for $t\ge2$, where $a$ is a constant.
   (i) Find the value of $a$ for which the graph of $M$ has no break at $t=2$.
   (ii) Hence find $M(5)$.
   (iii) Find the time at which $M$ first reaches 400.

   [6 marks]

   Total for question 4: 12 marks

End of questions