2.5 Modelling with functionsIB Maths: Applications and Interpretation SL: Subtopic test

1. 1
   A mobile phone plan costs 45 AED per month and includes 5 GB of data. Each extra GB used beyond 5 GB costs 8 AED. The monthly cost is $C$ AED when $d$ GB are used, $d\geq 0$.

   (a)

   Find the cost of using 9 GB in a month.

   [1 mark]

   • A$117$ AED
   • B$77$ AED
   • C$85$ AED
   • D$53$ AED

   (b)

   In one month a customer pays 101 AED. Find the amount of data used.

   [1 mark]

   • A$2$ GB
   • B$7$ GB
   • C$12.6$ GB
   • D$12$ GB

   (c)

   Write down the gradient of the model for $d>5$ and interpret it in context.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A footballer kicks a ball from the ground. Its height $h$ metres, when it has travelled a horizontal distance $x$ metres, is modelled by $h(x)=-0.04x^2+0.96x$, for $x\geq 0$ while the ball is in the air.

   (a)

   Find the horizontal distance at which the ball reaches its maximum height.

   [1 mark]

   • A$12$ m
   • B$24$ m
   • C$6$ m
   • D$5.76$ m

   (b)

   Find the maximum height reached by the ball.

   [1 mark]

   • A$12$ m
   • B$11.52$ m
   • C$5.76$ m
   • D$24$ m

   (c)

   Use your GDC to find the horizontal distance at which the ball is at a height of 5 m on its way up.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A cup of coffee is poured at $t=0$ minutes in a room. Its temperature $T$ °C is modelled by $T(t)=22+68e^{-0.05t}$, for $t\geq 0$.

   (a)

   (i) Find the temperature of the coffee when it is poured.
   (ii) Write down the equation of the horizontal asymptote of the graph of $T$ and interpret it in context.

   [3 marks]

   (b)

   (i) Use your GDC to find the time taken for the coffee to cool to 50 °C.
   (ii) Find the temperature of the coffee after 3 hours and comment on whether this is reasonable.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The depth $d$ metres of water in a harbour, $t$ hours after midnight, is modelled by $d(t)=3\sin(30t^\circ)+7$, for $t\geq 0$.

   (a)

   (i) Write down the amplitude of the model and the equation of its principal axis.
   (ii) Find the period of the model.
   (iii) Find the depth of the water at 14:00.
   (iv) Use your GDC to find the first time after midnight at which the depth is 8.5 m.

   [6 marks]

   (b)

   A second harbour has depth $D(t)=a\sin(bt^\circ)+k$, with $a,b>0$, a maximum depth of 12 m, a minimum depth of 4 m and a period of 12.4 hours.
   (i) Find the values of $a$, $b$ and $k$.
   (ii) Find the depth of the water at $t=3$.
   (iii) A ship needs a depth of at least 6 m. Use your GDC to find, for how long in each 12.4-hour cycle, the depth is less than 6 m.

   [6 marks]

   Total for question 4: 12 marks

End of questions