2.6 Modelling skillsIB Maths: Applications and Interpretation SL: Subtopic test

1. 1
   A rooftop water tank holds at most 400 litres. A pump starts filling the tank at $t=0$ minutes, and the volume of water $V$ litres in the tank is modelled by $V(t)=40+25t$.

   (a)

   Which of the following is the most reasonable domain for the model?

   [1 mark]

   • A$t\geq0$
   • B$0\leq t\leq16$
   • C$0\leq t\leq14.4$
   • D$0\leq t\leq17.6$

   (b)

   What does the number 25 in the model represent?

   [1 mark]

   • AThe volume of water increases by 25 litres every minute
   • BThe volume of water in the tank when the pump starts
   • CThe number of minutes the tank takes to fill
   • DThe greatest volume of water the tank can hold

   (c)

   Find $V(20)$ and comment on the reasonableness of the model at $t=20$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A scientist measures the mass $M$ grams of a radioactive sample $t$ days after the start: 80 g at $t=0$, 40 g at $t=6$ and 20 g at $t=12$. She wants to fit a model to these data.

   (a)

   Which type of model is most appropriate for these data?

   [1 mark]

   • AA linear model, $M=mt+c$
   • BA quadratic model, $M=at^2+bt+c$
   • CA sinusoidal model, $M=a\sin(bt)+d$
   • DAn exponential decay model, $M=ka^t$

   (b)

   She uses the model $M(t)=ka^t$. Find the value of $a$.

   [1 mark]

   • A$0.5$
   • B$0.891$
   • C$0.794$
   • D$1.12$

   (c)

   Use the model $M(t)=80a^t$, with $a$ from part (b), to predict the mass of the sample after 20 days.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A Ferris wheel has its lowest point 2 m above the ground and its highest point 52 m above the ground, and it completes one revolution every 20 minutes. A passenger is level with the axle, and rising, at $t=0$ minutes. The passenger's height $h$ metres above the ground is modelled by $h(t)=a\sin(bt^\circ)+d$, where $a,b>0$.

   (a)

   Find the values of $a$, $b$ and $d$.

   [3 marks]

   (b)

   (i) Find the height of the passenger at $t=5$.
   (ii) Use your GDC to find for how long, in one revolution, the passenger is higher than 40 m above the ground.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The population $P$ thousand of a town, $t$ years after the year 2000, was recorded as 40 thousand in 2000, 52 thousand in 2010 and 58 thousand in 2020. A town planner models the population by $P(t)=at^2+bt+c$.

   (a)

   (i) Write down three equations in $a$, $b$ and $c$ using the recorded data.
   (ii) Use your GDC to solve the equations.
   (iii) State what the value of $c$ represents.
   (iv) Find the population the model predicts for 2015.

   [6 marks]

   (b)

   Use the model $P(t)=-0.03t^2+1.5t+40$.
   (i) Find the year in which the model predicts the largest population, and that population.
   (ii) Use your GDC to find the time at which the model predicts the population to be zero.
   (iii) Comment on the suitability of the model for predicting the population after 2020.

   [6 marks]

   Total for question 4: 12 marks

End of questions