1.02z Models in context: use functions in modelling

2 Alex is comparing the cost of mobile phone contracts. Contract \(\boldsymbol { A }\) has a set-up cost of \(\pounds 40\) and then costs 4 p per minute. Contract \(\boldsymbol { B }\) has no set-up cost, does not charge for the first 100 minutes and then costs 6 p per minute.

1. Find an expression for the cost of each of the contracts in terms of \(m\), where \(m\) is the number of minutes for which the phone is used and \(m > 100\).
2. Hence find the value of \(m\) for which both contracts would cost the same.

4 A species of animal is to be introduced onto a remote island. Their food will consist only of various plants that grow on the island. A zoologist proposes two possible models for estimating the population \(P\) after \(t\) years. The diagrams show these models as they apply to the first 20 years. \includegraphics[max width=\textwidth, alt={}, center]{1a0e0afb-81be-45d1-8c86-f98e508e9a49-05_725_606_406_242} \includegraphics[max width=\textwidth, alt={}, center]{1a0e0afb-81be-45d1-8c86-f98e508e9a49-05_714_593_413_968}

1. Without calculation, describe briefly how the rate of growth of \(P\) will vary for the first 20 years, according to each of these two models. The equation of the curve for model A is \(P = 20 + 1000 \mathrm { e } ^ { - \frac { ( t - 20 ) ^ { 2 } } { 100 } }\).
   The equation of the curve for model B is \(P = 20 + 1000 \left( 1 - \mathrm { e } ^ { - \frac { t } { 5 } } \right)\).
2. Describe the behaviour of \(P\) that is predicted for \(t > 20\)
   1. using model A,
   2. using model B . There is only a limited amount of food available on the island, and the zoologist assumes that the size of the population depends on the amount of food available and on no other external factors.
3. State what is suggested about the long-term food supply by
   1. model A,
   2. model B.

5 \includegraphics[max width=\textwidth, alt={}, center]{1a0e0afb-81be-45d1-8c86-f98e508e9a49-06_371_339_255_251} For a cone with base radius \(r\), height \(h\) and slant height \(l\), the following formulae are given.
Curved surface area, \(S = \pi r l\) Volume, \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\) A container is to be designed in the shape of an inverted cone with no lid. The base radius is \(r \mathrm {~m}\) and the volume is \(V \mathrm {~m} ^ { 3 }\). The area of the material to be used for the cone is \(4 \pi \mathrm {~m} ^ { 2 }\).

1. Show that \(V = \frac { 1 } { 3 } \pi \sqrt { 16 r ^ { 2 } - r ^ { 6 } }\).
2. In this question you must show detailed reasoning. It is given that \(V\) has a maximum value for a certain value of \(r\).
   Find the maximum value of \(V\), giving your answer correct to 3 significant figures.

4 The size, \(P\), of a population of a certain species of insect at time \(t\) months is modelled by the following formula. \(P = 5000 - 1000 \cos ( 30 t ) ^ { \circ }\)

1. Write down the maximum size of the population.
2. Write down the difference between the largest and smallest values of \(P\).
3. Without giving any numerical values, describe briefly the behaviour of the population over time.
4. Find the time taken for the population to return to its initial size for the first time.
5. Determine the time on the second occasion when \(P = 4500\). A scientist observes the population over a period of time. He notices that, although the population varies in a way similar to the way predicted by the model, the variations become smaller and smaller over time, and \(P\) converges to 5000 .
6. Suggest a change to the model that will take account of this observation.

3 A publisher has to choose the price at which to sell a certain new book. The total profit, \(\pounds t\), that the publisher will make depends on the price, \(\pounds p\). He decides to use a model that includes the following assumptions.

• If the price is low, many copies will be sold, but the profit on each copy sold will be small, and the total profit will be small.
• If the price is high, the profit on each copy sold will be high, but few copies will be sold, and the total profit will be small.

The graphs below show two possible models. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Explain how model A is inconsistent with one of the assumptions given above.
2. Given that the equation of the curve in model B is quadratic, show that this equation is of the form \(t = k \left( 12 p - p ^ { 2 } \right)\), and find the value of the constant \(k\).
3. The publisher needs to make a total profit of at least \(\pounds 6400\). Use the equation found in part (b) to find the range of values within which model B suggests that the price of the book must lie.
4. Comment briefly on how realistic model B may be in the following cases.

1. In 1997 the average \(\mathrm { CO } _ { 2 }\) emissions of new cars in the UK was \(190 \mathrm {~g} / \mathrm { km }\).

In 2005 the average \(\mathrm { CO } _ { 2 }\) emissions of new cars in the UK had fallen to \(169 \mathrm {~g} / \mathrm { km }\).
Given \(\mathrm { Ag } / \mathrm { km }\) is the average \(\mathrm { CO } _ { 2 }\) emissions of new cars in the UK \(n\) years after 1997 and using a linear model,

1. form an equation linking \(A\) with \(n\). In 2016 the average \(\mathrm { CO } _ { 2 }\) emissions of new cars in the UK was \(120 \mathrm {~g} / \mathrm { km }\).
2. Comment on the suitability of your model in light of this information.

1. The temperature, \(\theta ^ { \circ } \mathrm { C }\), of a cup of tea \(t\) minutes after it was placed on a table in a room, is modelled by the equation

$$\theta = 18 + 65 \mathrm { e } ^ { - \frac { t } { 8 } } \quad t \geqslant 0$$ Find, according to the model,

1. the temperature of the cup of tea when it was placed on the table,
2. the value of \(t\), to one decimal place, when the temperature of the cup of tea was \(35 ^ { \circ } \mathrm { C }\).
3. Explain why, according to this model, the temperature of the cup of tea could not fall to \(15 ^ { \circ } \mathrm { C }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{bcbd842f-b2e2-4587-ab4c-15a57a449e5d-16_675_951_973_573} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} The temperature, \(\mu ^ { \circ } \mathrm { C }\), of a second cup of tea \(t\) minutes after it was placed on a table in a different room, is modelled by the equation $$\mu = A + B \mathrm { e } ^ { - \frac { t } { 8 } } \quad t \geqslant 0$$ where \(A\) and \(B\) are constants.
   Figure 2 shows a sketch of \(\mu\) against \(t\) with two data points that lie on the curve.
   The line \(l\), also shown on Figure 2, is the asymptote to the curve.
   Using the equation of this model and the information given in Figure 2
4. find an equation for the asymptote \(l\).

1. An advertising agency is monitoring the number of views of an online advert.

The equation $$\log _ { 10 } V = 0.072 t + 2.379 \quad 1 \leqslant t \leqslant 30 , t \in \mathbb { N }$$ is used to model the total number of views of the advert, \(V\), in the first \(t\) days after the advert went live.

1. Show that \(V = a b ^ { t }\) where \(a\) and \(b\) are constants to be found. Give the value of \(a\) to the nearest whole number and give the value of \(b\) to 3 significant figures.
2. Interpret, with reference to the model, the value of \(a b\). Using this model, calculate
3. the total number of views of the advert in the first 20 days after the advert went live. Give your answer to 2 significant figures.

1. The mass, \(A\) kg, of algae in a small pond, is modelled by the equation

$$A = p q ^ { t }$$ where \(p\) and \(q\) are constants and \(t\) is the number of weeks after the mass of algae was first recorded. Data recorded indicates that there is a linear relationship between \(t\) and \(\log _ { 10 } A\) given by the equation $$\log _ { 10 } A = 0.03 t + 0.5$$

1. Use this relationship to find a complete equation for the model in the form $$A = p q ^ { t }$$ giving the value of \(p\) and the value of \(q\) each to 4 significant figures.
2. With reference to the model, interpret
   1. the value of the constant \(p\),
   2. the value of the constant \(q\).
3. Find, according to the model,
   1. the mass of algae in the pond when \(t = 8\), giving your answer to the nearest 0.5 kg ,
   2. the number of weeks it takes for the mass of algae in the pond to reach 4 kg .
4. State one reason why this may not be a realistic model in the long term.

1. In this question you must show all stages of your working.

\section*{Solutions relying entirely on calculator technology are not acceptable.} The air pressure, \(P \mathrm {~kg} / \mathrm { cm } ^ { 2 }\), inside a car tyre, \(t\) minutes from the instant when the tyre developed a puncture is given by the equation $$P = k + 1.4 \mathrm { e } ^ { - 0.5 t } \quad t \in \mathbb { R } \quad t \geqslant 0$$ where \(k\) is a constant.
Given that the initial air pressure inside the tyre was \(2.2 \mathrm {~kg} / \mathrm { cm } ^ { 2 }\)

1. state the value of \(k\). From the instant when the tyre developed the puncture,
2. find the time taken for the air pressure to fall to \(1 \mathrm {~kg} / \mathrm { cm } ^ { 2 }\) Give your answer in minutes to one decimal place.
3. Find the rate at which the air pressure in the tyre is decreasing exactly 2 minutes from the instant when the tyre developed the puncture.
   Give your answer in \(\mathrm { kg } / \mathrm { cm } ^ { 2 }\) per minute to 3 significant figures.

1. A company makes drinks containers out of metal.

The containers are modelled as closed cylinders with base radius \(r \mathrm {~cm}\) and height \(h \mathrm {~cm}\) and the capacity of each container is \(355 \mathrm {~cm} ^ { 3 }\) The metal used

• for the circular base and the curved side costs 0.04 pence/ \(\mathrm { cm } ^ { 2 }\)
• for the circular top costs 0.09 pence/ \(\mathrm { cm } ^ { 2 }\)

Both metals used are of negligible thickness.

1. Show that the total cost, \(C\) pence, of the metal for one container is given by $$C = 0.13 \pi r ^ { 2 } + \frac { 28.4 } { r }$$
2. Use calculus to find the value of \(r\) for which \(C\) is a minimum, giving your answer to 3 significant figures.
3. Using \(\frac { \mathrm { d } ^ { 2 } C } { \mathrm {~d} r ^ { 2 } }\) prove that the cost is minimised for the value of \(r\) found in part (b).
4. Hence find the minimum value of \(C\), giving your answer to the nearest integer.

1. The distance a particular car can travel in a journey starting with a full tank of fuel was investigated.

• From a full tank of fuel, 40 litres remained in the car's fuel tank after the car had travelled 80 km
• From a full tank of fuel, 25 litres remained in the car's fuel tank after the car had travelled 200 km

Using a linear model, with \(V\) litres being the volume of fuel remaining in the car's fuel tank and \(d \mathrm {~km}\) being the distance the car had travelled,

1. find an equation linking \(V\) with \(d\). Given that, on a particular journey
   • the fuel tank of the car was initially full
   • the car continued until it ran out of fuel
     find, according to the model,
   • 1. the initial volume of fuel that was in the fuel tank of the car,
     2. the distance that the car travelled on this journey.
   In fact the car travelled 320 km on this journey.
2. Evaluate the model in light of this information.

12. \begin{figure}[h] \end{figure} Figure 5 shows the plan view of the design for a swimming pool.
The pool is modelled as a quarter of a circle joined to two equal sized rectangles as shown. Given that

• the quarter circle has radius \(x\) metres
• the rectangles each have length \(x\) metres and width \(y\) metres
• the total surface area of the swimming pool is \(100 \mathrm {~m} ^ { 2 }\)
  1. show that, according to the model, the perimeter \(P\) metres of the swimming pool is given by

$$P = 2 x + \frac { 200 } { x }$$

Use calculus to find the value of \(x\) for which \(P\) has a stationary value.

Prove, by further calculus, that this value of \(x\) gives a minimum value for \(P\) Access to the pool is by side \(A B\) shown in Figure 5.
Given that \(A B\) must be at least one metre,

determine, according to the model, whether the swimming pool with the minimum perimeter would be suitable.

5. A scientist is studying a population of lizards on an island and uses the linear model \(P = a + b t\) to predict the future population of the lizard where \(P\) is the population and \(t\) is the time in years after the start of the study. Given that

• The number of population was 900 , exactly 5 years after the start of the study.
• The number of population was 1200 , exactly 8 years after the start of the study.
  a. find a complete equation for the model.
  b. Sketch the graph of the population for the first 10 years.
  c. Suggest one criticism of this model.

1. In a simple model, the value, \(\pounds V\), of a car depends on its age, \(t\), in years.

The following information is available for \(\operatorname { car } A\)

• its value when new is \(\pounds 20000\)
• its value after one year is \(\pounds 16000\)
  1. Use an exponential model to form, for car \(A\), a possible equation linking \(V\) with \(t\).

The value of car \(A\) is monitored over a 10-year period.
Its value after 10 years is \(\pounds 2000\)

Evaluate the reliability of your model in light of this information. The following information is available for car \(B\)

• it has the same value, when new, as car \(A\)
• its value depreciates more slowly than that of \(\operatorname { car } A\)
• Explain how you would adapt the equation found in (a) so that it could be used to model the value of car \(B\).

12. $$\mathrm { f } ( x ) = 10 \mathrm { e } ^ { - 0.25 x } \sin x , \quad x \geqslant 0$$

1. Show that the \(x\) coordinates of the turning points of the curve with equation \(y = \mathrm { f } ( x )\) satisfy the equation \(\tan x = 4\) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{91a2f26a-add2-4b58-997d-2ae229548217-34_687_1029_495_518} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} Figure 3 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\).
2. Sketch the graph of \(H\) against \(t\) where $$\mathrm { H } ( t ) = \left| 10 \mathrm { e } ^ { - 0.25 t } \sin t \right| \quad t \geqslant 0$$ showing the long-term behaviour of this curve. The function \(\mathrm { H } ( t )\) is used to model the height, in metres, of a ball above the ground \(t\) seconds after it has been kicked. Using this model, find
3. the maximum height of the ball above the ground between the first and second bounce.
4. Explain why this model should not be used to predict the time of each bounce.

1. The height, \(h\) metres, of a tree, \(t\) years after being planted, is modelled by the equation

$$h ^ { 2 } = a t + b \quad 0 \leqslant t < 25$$ where \(a\) and \(b\) are constants.
Given that

• the height of the tree was 2.60 m , exactly 2 years after being planted
• the height of the tree was 5.10 m , exactly 10 years after being planted
  1. find a complete equation for the model, giving the values of \(a\) and \(b\) to 3 significant figures.

Given that the height of the tree was 7 m , exactly 20 years after being planted

evaluate the model, giving reasons for your answer.

11. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with equation $$y = ( \ln x ) ^ { 2 } \quad x > 0$$ The finite region \(R\), shown shaded in Figure 2, is bounded by the curve, the line with equation \(x = 2\), the \(x\)-axis and the line with equation \(x = 4\) The table below shows corresponding values of \(x\) and \(y\), with the values of \(y\) given to 4 decimal places.

1. Use the trapezium rule, with all the values of \(y\) in the table, to obtain an estimate for the area of \(R\), giving your answer to 3 significant figures.
2. Use algebraic integration to find the exact area of \(R\), giving your answer in the form $$y = a ( \ln 2 ) ^ { 2 } + b \ln 2 + c$$ where \(a\), \(b\) and \(c\) are integers to be found.

5. A curve \(C\) has parametric equations $$x = 2 t - 1 , \quad y = 4 t - 7 + \frac { 3 } { t } , \quad t \neq 0$$ Show that the Cartesian equation of the curve \(C\) can be written in the form $$y = \frac { 2 x ^ { 2 } + a x + b } { x + 1 } , \quad x \neq - 1$$ where \(a\) and \(b\) are integers to be found.

11. An archer shoots an arrow. The height, \(H\) metres, of the arrow above the ground is modelled by the formula $$H = 1.8 + 0.4 d - 0.002 d ^ { 2 } , \quad d \geqslant 0$$ where \(d\) is the horizontal distance of the arrow from the archer, measured in metres.
Given that the arrow travels in a vertical plane until it hits the ground,

1. find the horizontal distance travelled by the arrow, as given by this model.
2. With reference to the model, interpret the significance of the constant 1.8 in the formula.
3. Write \(1.8 + 0.4 d - 0.002 d ^ { 2 }\) in the form $$A - B ( d - C ) ^ { 2 }$$ where \(A , B\) and \(C\) are constants to be found. It is decided that the model should be adapted for a different archer.
   The adapted formula for this archer is $$H = 2.1 + 0.4 d - 0.002 d ^ { 2 } , \quad d \geqslant 0$$ Hence or otherwise, find, for the adapted model
4. 1. the maximum height of the arrow above the ground.
   2. the horizontal distance, from the archer, of the arrow when it is at its maximum height.

1. In a controlled experiment, the number of microbes, \(N\), present in a culture \(T\) days after the start of the experiment were counted. \(N\) and \(T\) are expected to satisfy a relationship of the form

$$N = a T ^ { b } , \quad \text { where } a \text { and } b \text { are constants }$$

1. Show that this relationship can be expressed in the form $$\log _ { 10 } N = m \log _ { 10 } T + c$$ giving \(m\) and \(c\) in terms of the constants \(a\) and/or \(b\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f7994129-07ee-4f6d-9531-08a15a38b794-18_1232_1046_804_513} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} Figure 3 shows the line of best fit for values of \(\log _ { 10 } N\) plotted against values of \(\log _ { 10 } T\)
2. Use the information provided to estimate the number of microbes present in the culture 3 days after the start of the experiment.
3. Explain why the information provided could not reliably be used to estimate the day when the number of microbes in the culture first exceeds 1000000 .
4. With reference to the model, interpret the value of the constant \(a\).

13. (a) Express \(2 \sin \theta - 1.5 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) State the value of \(R\) and give the value of \(\alpha\) to 4 decimal places. Tom models the depth of water, \(D\) metres, at Southview harbour on 18th October 2017 by the formula $$D = 6 + 2 \sin \left( \frac { 4 \pi t } { 25 } \right) - 1.5 \cos \left( \frac { 4 \pi t } { 25 } \right) , \quad 0 \leqslant t \leqslant 24$$ where \(t\) is the time, in hours, after 00:00 hours on 18th October 2017. Use Tom's model to
(b) find the depth of water at 00:00 hours on 18th October 2017,
(c) find the maximum depth of water,
(d) find the time, in the afternoon, when the maximum depth of water occurs. Give your answer to the nearest minute. Tom's model is supported by measurements of \(D\) taken at regular intervals on 18th October 2017. Jolene attempts to use a similar model in order to model the depth of water at Southview harbour on 19th October 2017. Jolene models the depth of water, \(H\) metres, at Southview harbour on 19th October 2017 by the formula $$H = 6 + 2 \sin \left( \frac { 4 \pi x } { 25 } \right) - 1.5 \cos \left( \frac { 4 \pi x } { 25 } \right) , \quad 0 \leqslant x \leqslant 24$$ where \(x\) is the time, in hours, after 00:00 hours on 19th October 2017.
By considering the depth of water at 00:00 hours on 19th October 2017 for both models,
(e) (i) explain why Jolene's model is not correct,
(ii) hence find a suitable model for \(H\) in terms of \(x\).

8. \begin{figure}[h] \end{figure} Figure 1 is a graph showing the trajectory of a rugby ball. The height of the ball above the ground, \(H\) metres, has been plotted against the horizontal distance, \(x\) metres, measured from the point where the ball was kicked. The ball travels in a vertical plane. The ball reaches a maximum height of 12 metres and hits the ground at a point 40 metres from where it was kicked.

1. Find a quadratic equation linking \(H\) with \(x\) that models this situation. The ball passes over the horizontal bar of a set of rugby posts that is perpendicular to the path of the ball. The bar is 3 metres above the ground.
2. Use your equation to find the greatest horizontal distance of the bar from \(O\).
3. Give one limitation of the model.

1. A scientist is studying a population of mice on an island.

The number of mice, \(N\), in the population, \(t\) months after the start of the study, is modelled by the equation $$N = \frac { 900 } { 3 + 7 \mathrm { e } ^ { - 0.25 t } } , \quad t \in \mathbb { R } , \quad t \geqslant 0$$

1. Find the number of mice in the population at the start of the study.
2. Show that the rate of growth \(\frac { \mathrm { d } N } { \mathrm {~d} t }\) is given by \(\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { N ( 300 - N ) } { 1200 }\) The rate of growth is a maximum after \(T\) months.
3. Find, according to the model, the value of \(T\). According to the model, the maximum number of mice on the island is \(P\).
4. State the value of \(P\).

1. A small factory makes bars of soap.

On any day, the total cost to the factory, \(\pounds y\), of making \(x\) bars of soap is modelled to be the sum of two separate elements:

• a fixed cost
• a cost that is proportional to the number of bars of soap that are made that day
  1. Write down a general equation linking \(y\) with \(x\), for this model.

The bars of soap are sold for \(\pounds 2\) each.
On a day when 800 bars of soap are made and sold, the factory makes a profit of £500 On a day when 300 bars of soap are made and sold, the factory makes a loss of \(\pounds 80\) Using the above information,

show that \(y = 0.84 x + 428\)

With reference to the model, interpret the significance of the value 0.84 in the equation. Assuming that each bar of soap is sold on the day it is made,

find the least number of bars of soap that must be made on any given day for the factory to make a profit that day.