2.4 Key features of graphsIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   A farmer uses 60 m of fencing to make a rectangular pen against a straight wall, so only three sides need fencing. The width of the pen, perpendicular to the wall, is $x$ metres, and its area is $A(x)=x(60-2x)$ m$^2$.

   (a)

   Find the equation of the axis of symmetry of the graph of $A$.

   [1 mark]

   • A$x=15$
   • B$x=30$
   • C$x=7.5$
   • D$x=450$

   (b)

   Use your GDC to find the maximum area of the pen.

   [1 mark]

   • A$15$ m$^2$
   • B$450$ m$^2$
   • C$900$ m$^2$
   • D$60$ m$^2$

   (c)

   Write down the zeros of $A$, and explain what the larger zero means in this context.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A school hires a coach for a trip. The cost per student, $C$ AED, is modelled by $C(n)=\frac{600}{n}+15$, where $n\ge1$ is the number of students on the trip.

   (a)

   Write down the equation of the horizontal asymptote of the graph of $C$.

   [1 mark]

   • A$n=0$
   • B$C=0$
   • C$C=600$
   • D$C=15$

   (b)

   Find the number of students for which the cost per student is exactly 25 AED.

   [1 mark]

   • A$24$
   • B$40$
   • C$60$
   • D$585$

   (c)

   Explain what the horizontal asymptote means in this context.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Two colonies of bacteria are grown in a laboratory. For $0\le t\le12$ days, the number of bacteria in colony A is $P(t)=500\times1.3^t$ and the number in colony B is $Q(t)=2000+300t$. Use your GDC where appropriate.

   (a)

   (i) Write down the number of bacteria in each colony at $t=0$.
   (ii) Find the value of $t$, for $t>0$, at which the two colonies have the same number of bacteria.

   [3 marks]

   (b)

   For $0\le t\le8.39$, colony B has more bacteria than colony A. Use your GDC to find the greatest difference $Q(t)-P(t)$ in this period, and the value of $t$ at which it occurs. Interpret your answers.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The temperature $T$ °C in a greenhouse $t$ hours after 06:00 is modelled by $T(t)=-0.1t^3+1.8t^2-5.4t+14$, for $0\le t\le12$. Use your GDC where appropriate.

   (a)

   (i) Use your GDC to find the coordinates of the local minimum point and the local maximum point of the graph of $T$.
   (ii) Find the greatest temperature in the greenhouse during the 12 hours, and justify your answer.
   (iii) Write down the lowest temperature in the greenhouse during the 12 hours.

   [6 marks]

   (b)

   (i) Use your GDC to find the values of $t$ for which $T(t)=12$.
   (ii) A heater switches on whenever the temperature is below 12 °C. Find the total time, in hours and minutes to the nearest minute, for which the heater is on.
   (iii) A fan switches on when the temperature first reaches 30 °C. Find the time, to the nearest minute, at which the fan switches on.

   [6 marks]

   Total for question 4: 12 marks

End of questions