19 GCE A level Pure Maths modelling questions PDF

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GCE (2017) AS and A level Mathematics 76 modelling problems

A collection of questions taken from the last 20 years of A level Mathematics papers – essentially every question with the word “model” included. Every question could be enhanced by the following: (a) Discuss the limitations of the model; (b) Discuss any refinements you might make to the model; Mark schemes can be found in the accompanying document on the emporium.

1. A cup of hot tea was placed on a table. At time t minutes after the cup was placed on the table, the temperature of the tea in the cup, θ °C, is modelled by the equation θ = 25 + Ae–0.03t where A is a constant. The temperature of the tea was 75 °C when the cup was placed on the table. (a) Find a complete equation for the model. (1) (b) Use the model to find the time taken for the tea to cool from 75 °C to 60 °C, giving your answer in minutes to one decimal place. (2) Two hours after the cup was placed on the table, the temperature of the tea was measured as 20.3 °C. Using this information, (c) evaluate the model, explaining your reasoning. (1) (Total for Question 1 is 4 marks) ___________________________________________________________________________ 2.

Rabbits were introduced onto an island. The number of rabbits, P, t years after they were introduced is modelled by the equation 1

P = 80e 5 t , t  ℝ,

t  0.

(a) Write down the number of rabbits that were introduced to the island. (1) (b) Find the number of years it would take for the number of rabbits to first exceed 1000. (2) dP (c) Find . dt

(2) dP = 50. (d) Find P when dt

(3) ___________________________________________________________________________

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3.

The speed, v m s1, of a lorry at time t seconds is modelled by v = 5(e0.1t – 1) sin (0.1t), 0  t  30. (a) Copy and complete the following table, showing the speed of the lorry at 5 second intervals. Use radian measure for 0.1t and give your values of v to 2 decimal places where appropriate. t v

0

5

10

15

1.56

7.23

17.36

20

25

(3) (b) Verify that, according to this model, the lorry is moving more slowly at t = 25 than at t = 24.5. (1) The distance, s metres, travelled by the lorry during the first 25 seconds is given by 25

 s =  v dt . 0

(c) Estimate s by using the trapezium rule with all the values from your table. (4) ___________________________________________________________________________ 4.

In the first month after opening, a mobile phone shop sold 280 phones. A model for future trading assumes that sales will increase by x phones per month for the next 35 months, so that (280 + x) phones will be sold in the second month, (280 + 2x) in the third month, and so on. Using this model with x = 5, calculate (a)

(i)

the number of phones sold in the 36th month, (2)

(ii)

the total number of phones sold over the 36 months. (2)

The shop sets a sales target of 17 000 phones to be sold over the 36 months. Using the same model, (b)

find the least value of x required to achieve this target.

(4) ___________________________________________________________________________

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5.

The value of a car is modelled by the formula V = 16 000e−kt + A,

t  0, t ∈ ℝ

where V is the value of the car in pounds, t is the age of the car in years, and k and A are positive constants. Given that the value of the car is £17 500 when new and £13 500 two years later, (a) find the value of A, (1)  2  (b) show that k = ln    3 (4) (c) Find the age of the car, in years, when the value of the car is £6000. Give your answer to 2 decimal places. (4) (Total 9 marks) ___________________________________________________________________________ 6.

(a) An arithmetic series has first term a and common difference d. Prove that the sum of the first n terms of the series is 1 2

n[2a + (n – 1)d]. (4)

A company made a profit of £54 000 in the year 2001. A model for future performance assumes that yearly profits will increase in an arithmetic sequence with common difference £d. This model predicts total profits of £619 200 for the 9 years 2001 to 2009 inclusive. (b) Find the value of d. (4) Using your value of d, (c) find the predicted profit for the year 2011. (2) An alternative model assumes that the company’s yearly profits will increase in a geometric sequence with common ratio 1.06. Using this alternative model and again taking the profit in 2001 to be £54 000, (d) find the predicted profit for the year 2011. (3) ___________________________________________________________________________

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7.

A tree was planted in the ground. Its height, H metres, was measured t years after planting. Exactly 3 years after planting, the height of the tree was 2.35 metres. Exactly 6 years after planting, the height of the tree was 3.28 metres. Using a linear model, (a) find an equation linking H with t. (3) The height of the tree was approximately 140 cm when it was planted. (b) Explain whether or not this fact supports the use of the linear model in part (a). (2) (Total for Question 7 is 5 marks) ___________________________________________________________________________

8.

The rate of decay of the mass of a particular substance is modelled by the differential equation dx 5  x, t  0 , dt 2

where x is the mass of the substance measured in grams and t is the time measured in days. Given that x = 60 when t = 0, (a) solve the differential equation, giving x in terms of t. You should show all steps in your working and give your answer in its simplest form. (4) (b) Find the time taken for the mass of the substance to decay from 60 grams to 20 grams. Give your answer to the nearest minute. (3) (Total 7 marks) ___________________________________________________________________________

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9.

Water is being heated in an electric kettle. The temperature,  °C, of the water t seconds after the kettle is switched on, is modelled by the equation

 = 120 – 100e– t,

0  t  T.

(a) State the value of  when t = 0. (1) Given that the temperature of the water in the kettle is 70 °C when t = 40, (b) find the exact value of  , giving your answer in the form

ln a , where a and b are b

integers. (4) When t = T, the temperature of the water reaches 100 °C and the kettle switches off. (c) Calculate the value of T to the nearest whole number. (2) ___________________________________________________________________________

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10.

Joan brings a cup of hot tea into a room and places the cup on a table. At time t minutes after Joan places the cup on the table, the temperature,  °C, of the tea is modelled by the equation

 = 20 + Ae−kt, where A and k are positive constants. Given that the initial temperature of the tea was 90 °C, (a) find the value of A. (2) The tea takes 5 minutes to decrease in temperature from 90 °C to 55 °C. (b) Show that k =

1 ln 2. 5

(3) (c) Find the rate at which the temperature of the tea is decreasing at the instant when t = 10. Give your answer, in °C per minute, to 3 decimal places. (3) ___________________________________________________________________________

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11.

A study is being carried out on two colonies of ants. The number of ants NA in colony A, t years after the start of the study, is modelled by the equation

NA 3000  600e0.12t

t  , t 0

Using the model, (a) find the time taken, from the start of the study, for the number of ants in colony A to double. Give your answer, in years, to 2 decimal places. (5) dNA  pN A  q , where p and q are constants to be determined. (b) Show that dt

(3) The number of ants NB in colony B, t years after the start of the study, is modelled by the equation

NB 2900  Ce kt

t  , t 0

where C and k are positive constants. According to this model, there will be 3100 ants in colony B one year after the start of the study and 3400 ants in colony B two years after the start of the study. 5 (c) (i) Show that k ln    2

(ii) Find the value of C. (4) (Total 12 marks) ___________________________________________________________________________

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12.

Initially the number of fish in a lake is 500 000. The population is then modelled by the recurrence relation un + 1 = 1.05un – d, u0 = 500 000. In this relation un is the number of fish in the lake after n years and d is the number of fish which are caught each year. Given that d = 15 000, (a) calculate u1 , u2 and u3 and comment briefly on your results. (3) Given that d = 100 000, (b) show that the population of fish dies out during the sixth year. (3) (c) Find the value of d which would leave the population each year unchanged. (2) ___________________________________________________________________________

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13.

Figure 2 y 10

2

O

10

12

x

Figure 2 shows the cross-section of a road tunnel and its concrete surround. The curved section of the tunnel is modelled by the curve with equation y = 8

 x   , in the interval  sin 10  

0  x  10. The concrete surround is represented by the shaded area bounded by the curve, the x-axis and the lines x = 2, x = 12 and y = 10. The units on both axes are metres. (a) Using this model, copy and complete the table below, giving the values of y to 2 decimal places. x

0

2

y

0

6.13

4

6

8

10 0 (2)



The area of the cross-section of the tunnel is given by 

10

0

y dx .

(b) Estimate this area, using the trapezium rule with all the values from your table. (4) (c) Deduce an estimate of the cross-sectional area of the concrete surround. (1) (d) State, with a reason, whether your answer in part (c) over-estimates or under-estimates the true value. (2) ___________________________________________________________________________

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14.

On a journey, the average speed of a car is v m s1. For v  5, the cost per kilometre, C pence, of the journey is modelled by C=

160 v 2  . 100 v

Using this model, (a) show, by calculus, that there is a value of v for which C has a stationary value, and find this value of v. (5) (b) Justify that this value of v gives a minimum value of C. (2) (c) Find the minimum value of C and hence find the minimum cost of a 250 km car journey. (3) ___________________________________________________________________________

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15.

Figure 2

Figure 3

Figure 2 shows the entrance to a road tunnel. The maximum height of the tunnel is measured as 5 metres and the width of the base of the tunnel is measured as 6 metres. Figure 3 shows a quadratic curve BCA used to model this entrance. The points A, O, B and C are assumed to lie in the same vertical plane and the ground AOB is assumed to be horizontal. (a) Find an equation for curve BCA. (3) A coach has height 4.1 m and width 2.4 m. (b) Determine whether or not it is possible for the coach to enter the tunnel. (2) (c) Suggest a reason why this model may not be suitable to determine whether or not the coach can pass through the tunnel. (1) (Total for Question 15 is 6 marks) ___________________________________________________________________________

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16.

A company plans to extract oil from an oil field. The daily volume of oil V, measured in barrels that the company will extract from this oil field depends upon the time, t years, after the start of drilling. The company decides to use a model to estimate the daily volume of oil that will be extracted. The model includes the following assumptions: • The initial daily volume of oil extracted from the oil field will be 16 000 barrels. • The daily volume of oil that will be extracted exactly 4 years after the start of drilling will be 9000 barrels. • The daily volume of oil extracted will decrease over time. The diagram below shows the graphs of two possible models. V

V

(0, 16 000)

(0, 16 000) (4, 9 000)

(4, 9 000)

O

Model A

O

t

Model B

t

(a) (i) Use model A to estimate the daily volume of oil that will be extracted exactly 3 years after the start of drilling. (ii) Write down a limitation of using model A. (2) (b) (i) Using an exponential model and the information given in the question, find a possible equation for model B. (ii) Using your answer to (b)(i) estimate the daily volume of oil that will be extracted exactly 3 years after the start of drilling. (5) (Total for Question 16 is 7 marks) ___________________________________________________________________________

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17.

The speed of a small jet aircraft was measured every 5 seconds, starting from the time it turned onto a runway, until the time when it left the ground. The results are given in the table below with the time in seconds and the speed in m s−1. Time (s)

0

5

10

15

20

25

Speed (m s−1)

2

5

10

18

28

42

Using all of this information, (a) estimate the length of runway used by the jet to take off. (3) Given that the jet accelerated smoothly in these 25 seconds, (b) explain whether your answer to part (a) is an underestimate or an overestimate of the length of runway used by the jet to take off. (1) (Total for Question 17 is 4 marks) ___________________________________________________________________________ 18.

The mass, m grams, of a radioactive substance t years after first being observed, is modelled by the equation m = 25e1–kt where k is a positive constant. (a) State the value of m when the radioactive substance was first observed. (1) Given that the mass is 50 grams, 10 years after first being observed, (b) show that k =

1 1  ln e 10  2 

(4) (c) Find the value of t when m = 20, giving your answer to the nearest year. (3) ___________________________________________________________________________

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19.

A population of deer is introduced into a park. The population P at t years after the deer have been introduced is modelled by P

2000a t , 4  at

where a is a constant. Given that there are 800 deer in the park after 6 years, (a) calculate, to 4 decimal places, the value of a, (4) (b) use the model to predict the number of years needed for the population of deer to increase from 800 to 1800. (4) (c) With reference to this model, give a reason why the population of deer cannot exceed 2000. (1) ___________________________________________________________________________ 20.

Water is being heated in a kettle. At time t seconds, the temperature of the water is θ °C. The rate of increase of the temperature of the water at any time t is modelled by the differential equation d = λ(120 – θ), dt

θ ≤ 100

where λ is a positive constant. Given that θ = 20 when t = 0, (a) solve this differential equation to show that θ = 120 – 100e–λt (8) When the temperature of the water reaches 100°C, the kettle switches off. (b) Given that λ = 0.01, find the time, to the nearest second, when the kettle switches off. (3) ___________________________________________________________________________

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21. A bacterial culture has area p mm2 at time t hours after the culture was placed onto a circular dish. A scientist states that at time t hours, the rate of increase of the area of the culture can be modelled as being proportional to the area of the culture. (a) Show that the scientist’s model for p leads to the equation p = aekt, where a and k are constants. (4) The scientist measures the values for p at regular intervals during the first 24 hours after the culture was placed onto the dish. She plots a graph of ln p against t and finds that the points on the graph lie close to a straight line with gradient 0.14 and vertical intercept 3.95. (b) Estimate, to 2 significant figures, the value of a and the value of k. (3) (c) Hence show that the model for p can be rewritten as p = abt, stating, to 3 significant figures, the value of the constant b. (2) With reference to this model, (d) (i) interpret the value of the constant a, (ii) interpret the value of the constant b. (2) (e) State a long term limitation of the model for p. (1) (Total for Question 21 is 12 marks) ___________________________________________________________________________

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22.

In a simple model, the value, £V, of a car depends on its age, t, in years. The following information is available for car A  

its value when new is £20 000 its value after one year is £16 000

(a) Use an exponential model to form, for car A, a possible equation linking V with t. (4) The value of car A is monitored over a 10-year period. Its value after 10 years is £2 000 (b) Evaluate the reliability of your model in light of this information. (2) 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 car A

(c) Explain how you would adapt the equation found in (a) so that it could be used to model the value of car B. (1) (Total for Question 22 is 7 marks) ___________________________________________________________________________

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23.

A small factory makes bars of soap. On any day, the total cost to the factory, £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.

(a) Write down a general equation linking y with x, for this model. (1) The bars of soap are sold for £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 £80. Using the above information, (b) show that y = 0.84x + 428 (3) (c) With reference to the model, interpret the significance of the value 0.84 in the equation. (1) Assuming that each bar of soap is sold on the day it is made, (d) 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. (2) (Total for Question 23 is 7 marks) ___________________________________________________________________________

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24.

The rate of increase of the number, N, of fish in a lake is modelled by the differential equation dN  kt  1  5000  N  ,  t dt

t > 0, 0 < N < 5000

In the given equation, the time t is measured in years from the start of January 2000 and k is a positive constant. (a) By solving the differential equation, show that N = 5000 – Ate–kt where A is a positive constant. (5) After one year, at the start of January 2001, there are 1200 fish in the lake. After two years, at the start of January 2002, there are 1800 fish in the lake. (b) Find the exact value of the constant A and the exact value of the constant k. (4) (c) Hence find the number of fish in the lake after five years. Give your answer to the nearest hundred fish....