1.02z Models in context: use functions in modelling

9 An analyst believes that the sales of a particular electronic device are growing exponentially. In 2015 the sales were 3.1 million devices and the rate of increase in the annual sales is 0.8 million devices per year.

1. Find a model to represent the annual sales, defining any variables used.
2. In 2017 the sales were 5.2 million devices. Determine whether this is consistent with the model in part (i).
3. The analyst uses the model in part (i) to predict the sales for 2025. Comment on the reliability of this prediction.

4. \begin{figure}[h] \end{figure} A manufacturer produces cartons for fruit juice. Each carton is in the shape of a closed cuboid with base dimensions \(2 x \mathrm {~cm}\) by \(x \mathrm {~cm}\) and height \(h \mathrm {~cm}\), as shown in Fig. 4. Given that the capacity of a carton has to be \(1030 \mathrm {~cm} ^ { 3 }\),

1. express \(h\) in terms of \(x\),
2. show that the surface area, \(A \mathrm {~cm} ^ { 2 }\), of a carton is given by $$A = 4 x ^ { 2 } + \frac { 3090 } { x } .$$

10 As part of an experiment, Zena puts a bucket of hot water outside on a day when the outside temperature is \(0 ^ { \circ } \mathrm { C }\). She measures the temperature of the water after 10 minutes and after 20 minutes. Her results are shown below. Zena models the relationship between \(\theta\), the temperature of the water in \({ } ^ { \circ } \mathrm { C }\), and \(t\), the time in minutes, by $$\theta = A \times 10 ^ { - k t }$$ where \(A\) and \(k\) are constants. 10

1. Using \(t = 0\), explain how the value of \(A\) relates to the experiment. 10
2. Show that $$\log _ { 10 } \theta = \log _ { 10 } A - k t$$ 10
3. Using Zena's results, calculate the values of \(A\) and \(k\).
   10
4. Zena states that the temperature of the water will be less than \(1 ^ { \circ } \mathrm { C }\) after 45 minutes. Determine whether the model supports this statement.
   10
5. Explain why Zena's model is unlikely to accurately give the value of \(\theta\) after 45 minutes.

6 A cup of tea is served at \(80 ^ { \circ } \mathrm { C }\) in a room which is kept at a constant \(20 ^ { \circ } \mathrm { C }\). The temperature, \(T ^ { \circ } \mathrm { C }\), of the tea after \(t\) minutes can be modelled by assuming that the rate of change of \(T\) is proportional to the difference in temperature between the tea and the room.

1. Explain why the rate of change of the temperature in this model is given by \(\frac { \mathrm { d } T } { \mathrm {~d} t } = - k ( T - 20 )\), where \(k\) is a positive constant.
2. Show by integration that the temperature of the tea after \(t\) minutes is given by \(T = 20 + 60 \mathrm { e } ^ { - k t }\).
3. After 2 minutes the tea has cooled to \(60 ^ { \circ } \mathrm { C }\). Find the value of \(k\).

The curve \(C\) has equation $$y = \frac{5x^2 + 5x + 1}{x^2 + x + 1}.$$

1. Find the equation of the asymptote of \(C\). [2]
2. Show that, for all real values of \(x\), \(-\frac{1}{5} \leqslant y < 5\). [4]
3. Find the coordinates of any stationary points of \(C\). [2]
4. Sketch \(C\), stating the coordinates of any intersections with the \(y\)-axis. [2]

\includegraphics{figure_4} A manufacturer produces cartons for fruit juice. Each carton is in the shape of a closed cuboid with base dimensions \(2x\) cm by \(x\) cm and height \(h\) cm, as shown in Fig. 4. Given that the capacity of a carton has to be \(1030\) cm³,

1. express \(h\) in terms of \(x\), [2]
2. show that the surface area, \(A\) cm², of a carton is given by $$A = 4x^2 + \frac{3090}{x}.$$ [3]

A container made from thin metal is in the shape of a right circular cylinder with height \(h\) cm and base radius \(r\) cm. The container has no lid. When full of water, the container holds 500 cm³ of water. Show that the exterior surface area, \(A\) cm², of the container is given by $$A = \pi r^2 + \frac{1000}{r}.$$ [4]

\includegraphics{figure_4} A manufacturer produces cartons for fruit juice. Each carton is in the shape of a closed cuboid with base dimensions 2\(x\) cm by \(x\) cm and height \(h\) cm, as shown in Fig. 4. Given that the capacity of a carton has to be 1030 cm\(^3\),

1. express \(h\) in terms of \(x\), [2]
2. show that the surface area, \(A\) cm\(^2\), of a carton is given by $$A = 4x^2 + \frac{3090}{x}.$$ [3]

A cuboid has a volume of \(8 \text{m}^3\). The base of the cuboid is square with sides of length \(x\) metres. The surface area of the cuboid is \(A \text{m}^2\).

1. Show that \(A = 2x^2 + \frac{32}{x}\). [3]
2. Find \(\frac{dA}{dx}\). [3]
3. Find the value of \(x\) which gives the smallest surface area of the cuboid, justifying your answer. [4]

A store begins to stock a new range of DVD players and achieves sales of £1500 of these products during the first month. In a model it is assumed that sales will decrease by £\(x\) in each subsequent month, so that sales of £\((1500 - x)\) and £\((1500 - 2x)\) will be achieved in the second and third months respectively. Given that sales total £8100 during the first six months, use the model to

1. find the value of \(x\), [4]
2. find the expected value of sales in the eighth month, [2]
3. show that the expected total of sales in pounds during the first \(n\) months is given by \(kn(51 - n)\), where \(k\) is an integer to be found. [3]
4. Explain why this model cannot be valid over a long period of time. [1]

\includegraphics{figure_2} A rectangular sheet of metal measures 50 cm by 40 cm. Squares of side \(x\) cm are cut from each corner of the sheet and the remainder is folded along the dotted lines to make an open tray, as shown in Fig. 2.

1. Show that the volume, \(V\) cm\(^3\), of the tray is given by $$V = 4x(x^2 - 45x + 500)$$ [3]
2. State the range of possible values of \(x\). [1]
3. Find the value of \(x\) for which \(V\) is a maximum. [4]
4. Hence find the maximum value of \(V\). [2]
5. Justify that the value of \(V\) you found in part (d) is a maximum. [2]

\includegraphics{figure_3} A manufacturer produces cartons for fruit juice. Each carton is in the shape of a closed cuboid with base dimensions \(2x\) cm by \(x\) cm and height \(h\) cm, as shown in Fig. 3. Given that the capacity of a carton has to be 1030 cm\(^3\),

1. express \(h\) in terms of \(x\), [2]
2. show that the surface area, \(A\) cm\(^2\), of a carton is given by \(A = 4x^2 + \frac{3090}{x}\). [3]

The manufacturer needs to minimise the surface area of a carton.

1. Use calculus to find the value of \(x\) for which \(A\) is a minimum. [5]
2. Calculate the minimum value of \(A\). [2]
3. Prove that this value of \(A\) is a minimum. [2]

\includegraphics{figure_3} A rectangular sheet of metal measures 50 cm by 40 cm. Squares of side \(x\) cm are cut from each corner of the sheet and the remainder is folded along the dotted lines to make an open tray, as shown in Fig. 3.

1. Show that the volume, \(V\) cm\(^3\), of the tray is given by \(V = 4x(x^2 - 45x + 500)\). [3]
2. State the range of possible values of \(x\). [1]
3. Find the value of \(x\) for which \(V\) is a maximum. [4]
4. Hence find the maximum value of \(V\). [2]
5. Justify that the value of \(V\) you found in part (d) is a maximum. [2]

The function f is defined by $$\text{f}(x) \equiv 3 - x^2, \quad x \in \mathbb{R}, \quad x \geq 0.$$

1. State the range of f. [1]
2. Sketch the graphs of \(y = \text{f}(x)\) and \(y = \text{f}^{-1}(x)\) on the same diagram. [3]
3. Find an expression for f\(^{-1}(x)\) and state its domain. [3]

The function g is defined by $$\text{g}(x) \equiv \frac{8}{3-x}, \quad x \in \mathbb{R}, \quad x \neq 3.$$

1. Evaluate fg\((-3)\). [2]
2. Solve the equation $$\text{f}^{-1}(x) = \text{g}(x).$$ [3]

On 18 March 2019 there were 12 hours of daylight in Inverness. On 16 June 2019, 90 days later, there will be 18 hours of daylight in Inverness. Jude decides to model the number of hours of daylight in Inverness, \(N\), by the formula $$N = A + B\sin t°$$ where \(t\) is the number of days after 18 March 2019.

1. 1. State the value that Jude should use for \(A\). [1 mark]
   2. State the value that Jude should use for \(B\). [1 mark]
   3. Using Jude's model, calculate the number of hours of daylight in Inverness on 15 May 2019, 58 days after 18 March 2019. [1 mark]
   4. Using Jude's model, find how many days during 2019 will have at least 17.4 hours of daylight in Inverness. [4 marks]
   5. Explain why Jude's model will become inaccurate for 2020 and future years. [1 mark]
2. Anisa decides to model the number of hours of daylight in Inverness with the formula $$N = A + B\sin \left(\frac{360}{365}t\right)°$$ Explain why Anisa's model is better than Jude's model. [1 mark]

Scientists observed a colony of seabirds over a period of 10 years starting in 2010. They concluded that the number of birds in the colony, its population \(P\), could be modelled by a formula of the form $$P = a(10^{bt})$$ where \(t\) is the time in years after 2010, and \(a\) and \(b\) are constants.

1. Explain what the value of \(a\) represents. [1 mark]
2. Show that \(\log_{10} P = bt + \log_{10} a\) [2 marks]
3. The table below contains some data collected by the scientists.

   Year	2013	2015
   \(t\)	3
   \(P\)	10200	12800
   \(\log_{10} P\)	4.0086

   1. Complete the table, giving the \(\log_{10} P\) value to 5 significant figures. [1 mark]
   2. Use the data to calculate the value of \(a\) and the value of \(b\). [4 marks]
   3. Use the model to estimate the population of the colony in 2024. [2 marks]
4. 1. State an assumption that must be made in using the model to estimate the population of the colony in 2024. [1 mark]
   2. Hence comment, with a reason, on the reliability of your estimate made in part (c)(iii). [1 mark]

An open-topped fish tank is to be made for an aquarium. It will have a square horizontal base, rectangular vertical sides and a volume of 60 m\(^3\) The materials cost:

• £15 per m\(^2\) for the base
• £8 per m\(^2\) for the sides.

1. Modelling the sides and base of the fish tank as laminae, use calculus to find the height of the tank for which the overall cost of the materials has its minimum value. Fully justify your answer. [8 marks]
2. 1. In reality, the thickness of the base and sides of the tank is 2.5 cm Briefly explain how you would refine your modelling to take account of the thickness of the sides and base of the tank. [1 mark]
   2. How would your refinement affect your answer to part (a)? [1 mark]

A market trader notices that daily sales are dependent on two variables: number of hours, \(t\), after the stall opens total sales, \(x\), in pounds since the stall opened. The trader models the rate of sales as directly proportional to \(\frac{8 - t}{x}\) After two hours the rate of sales is £72 per hour and total sales are £336

1. Show that $$x \frac{dx}{dt} = 4032(8 - t)$$ [3 marks]
2. Hence, show that $$x^2 = 4032t(16 - t)$$ [3 marks]
3. The stall opens at 09.30.
   1. The trader closes the stall when the rate of sales falls below £24 per hour. Using the results in parts (a) and (b), calculate the earliest time that the trader closes the stall. [6 marks]
   2. Explain why the model used by the trader is not valid at 09.30. [2 marks]

Helen is creating a mosaic pattern by placing square tiles next to each other along a straight line. \includegraphics{figure_9} The area of each tile is half the area of the previous tile, and the sides of the largest tile have length \(w\) centimetres.

1. Find, in terms of \(w\), the length of the sides of the second largest tile. [1 mark]
2. Assume the tiles are in contact with adjacent tiles, but do not overlap. Show that, no matter how many tiles are in the pattern, the total length of the series of tiles will be less than \(3.5w\). [4 marks]
3. Helen decides the pattern will look better if she leaves a 3 millimetre gap between adjacent tiles. Explain how you could refine the model used in part (b) to account for the 3 millimetre gap, and state how the total length of the series of tiles will be affected. [2 marks]

A student is conducting an experiment in a laboratory to investigate how quickly liquids cool to room temperature. A beaker containing a hot liquid at an initial temperature of \(75°C\) cools so that the temperature, \(\theta °C\), of the liquid at time \(t\) minutes can be modelled by the equation $$\theta = 5(4 + \lambda e^{-kt})$$ where \(\lambda\) and \(k\) are constants. After 2 minutes the temperature falls to \(68°C\).

1. Find the temperature of the liquid after 15 minutes. Give your answer to three significant figures. [7 marks]
2. 1. Find the room temperature of the laboratory, giving a reason for your answer. [2 marks]
   2. Find the time taken in minutes for the liquid to cool to \(1°C\) above the room temperature of the laboratory. [2 marks]
3. Explain why the model might need to be changed if the experiment was conducted in a different place. [1 mark]

\includegraphics{figure_4} Figure 4 shows the plan view of the design for a swimming pool. The shape of this pool \(ABCDEA\) consists of a rectangular section \(ABDE\) joined to a semicircular section \(BCD\) as shown in Figure 4. Given that \(AE = 2x\) metres, \(ED = y\) metres and the area of the pool is \(250\text{m}^2\),

1. show that the perimeter, \(P\) metres, of the pool is given by $$P = 2x + \frac{250}{x} + \frac{\pi x}{2}$$ [4]
2. Explain why \(0 < x < \sqrt{\frac{500}{\pi}}\) [2]
3. Find the minimum perimeter of the pool, giving your answer to \(3\) significant figures. [4]

\includegraphics{figure_1} A stone is thrown over level ground from the top of a tower, \(X\). The height, \(h\), in meters, of the stone above the ground level after \(t\) seconds is modelled by the function. $$h(t) = 7 + 21t - 4.9t^2, \quad t \geq 0$$ A sketch of \(h\) against \(t\) is shown in Figure 1. Using the model,

1. give a physical interpretation of the meaning of the constant term 7 in the model. [1]
2. find the time taken after the stone is thrown for it to reach ground level. [3]
3. Rearrange \(h(t)\) into the form \(A - B(t - C)^2\), where \(A\), \(B\) and \(C\) are constants to be found. [3]
4. Using your answer to part c or otherwise, find the maximum height of the stone above the ground, and the time after which this maximum height is reached. [2]

\includegraphics{figure_2} Figure 2 shows a solid cuboid \(ABCDEFGH\). \(AB = x\) cm, \(BC = 2x\) cm, \(AE = h\) cm The total surface area of the cuboid is 180 cm\(^2\). The volume of the cuboid is \(V\) cm\(^3\).

1. Show that \(V = 60x - \frac{4x^3}{3}\) [4]

Given that \(x\) can vary,

1. use calculus to find, to 3 significant figures, the value of \(x\) for which \(V\) is a maximum. Justify that this value of \(x\) gives a maximum value of \(V\). [5]
2. Find the maximum value of \(V\), giving your answer to the nearest cm\(^3\). [2]

In an experiment 500 fruit flies were released into a controlled environment. After 10 days there were 650 fruit flies present. Munirah believes that \(N\), the number of fruit flies present at time \(t\) days after the fruit flies are released, will increase at the rate of 4.4% per day. She proposes that the situation is modelled by the formula \(N = Ak^t\).

1. Write down the values of \(A\) and \(k\). [2]
2. Determine whether the model is consistent with the value of \(N\) at \(t = 10\). [2]
3. What does the model suggest about the number of fruit flies in the long run? [1]

Subsequently it is found that for large values of \(t\) the number of fruit flies in the controlled environment oscillates about 750. It is also found that as \(t\) increases the oscillations decrease in magnitude. Munirah proposes a second model in the light of this new information. $$N = 750 - 250 \times e^{-0.092t}$$

1. Identify three ways in which this second model is consistent with the known data. [3]
2. 1. Identify one feature which is not accounted for by the second model. [1]
   2. Give an example of a mathematical function which needs to be incorporated in the model to account for this feature. [1]

\includegraphics{figure_5} \includegraphics{figure_6} A suspension bridge cable \(PQR\) hangs between the tops of two vertical towers, \(AP\) and \(BR\), as shown in Figure 5. A walkway \(AOB\) runs between the bases of the towers, directly under the cable. The towers are 100 m apart and each tower is 24 m high. At the point \(O\), midway between the towers, the cable is 4 m above the walkway. The points \(P\), \(Q\), \(R\), \(A\), \(O\) and \(B\) are assumed to lie in the same vertical plane and \(AOB\) is assumed to be horizontal. Figure 6 shows a symmetric quadratic curve \(PQR\) used to model this cable. Given that \(O\) is the origin,

1. find an equation for the curve \(PQR\). [3] Lee can safely inspect the cable up to a height of 12 m above the walkway. A defect is reported on the cable at a location 19 m horizontally from one of the towers.
2. Determine whether, according to the model, Lee can safely inspect this defect. [2]