Modelling with Calculus 6992/2

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1 Centre Number Candidate Number For Examiner s Use Surname Other Names Candidate Signature Examiner s Initials Free-Standing Mathematics Qualification Advanced Level June 2015 Modelling with Calculus 6992/2 Unit 12 Friday 15 May am to am For this paper you must have: a clean copy of the Data Sheet (enclosed) a calculator a ruler. Question TOTAL Mark Time allowed 1 hour 30 minutes Instructions Use black ink or black ball-point pen. Pencil should only be used for drawing. Fill in the es at the top of this page. Answer all questions. Write the question part reference (eg (a), (b)(i) etc) in the left-hand margin. You must answer each question in the space provided for that question. If you require extra space, use an AQA supplementary answer book; do not use the space provided for a different question. around each page. Show all necessary working; otherwise marks for method may be lost. Do all rough work in this book. Cross through any work that you do not want to be marked. The final answer to questions requiring the use of tables or calculators should normally be given to three significant figures, unless stated otherwise. You may not refer to the copy of the Data Sheet that was available prior to this examination. A clean copy is enclosed for your use. Information The marks for questions are shown in brackets. The maximum mark for this paper is 60. You may use either a scientific calculator or a graphics calculator. Advice You do not necessarily need to use all the space provided. (JUN156992/201) P89764/Jun15/E3 6992/2

2 2 Section A Answer all questions. Answer each question in the space provided for that question. Use Baseball on page 2 of the Data Sheet. 1 In a baseball game, Mark hits a ball. The vertical height of the ball, y feet above A, the point at which it is hit, is given by y ¼ 64t 16t 2 where t is the time in seconds after the ball is hit. (a) Find the vertical height of the ball above A when t ¼ 3. [1 mark] (b) (c) Find dy dt. Find t when dy dt ¼ 0. (d) Hence find the maximum vertical height of the ball above A. (e) Mark hits the ball when it is 3 feet above the level of the horizontal ground. The ball hits the wall at a height of 7 feet above the ground and at a horizontal distance of 402 feet from A. Find the horizontal speed of the ball immediately after it is hit. [5 marks] Answer space for question 1 (02)

3 3 Answer space for question 1 Turn over s (03)

4 4 Answer space for question 1 (04)

5 5 Answer space for question 1 Turn over s (05)

6 6 Section B Answer all questions. Answer each question in the space provided for that question. Use Tablets on page 3 of the Data Sheet. 2 When Chloe prices the tablets so that she makes a profit of x on each tablet she sells, the number of tablets that she sells in a month may be modelled by x x 2 Thus, the profit, P, which Chloe makes, may be modelled by P ¼ 400x 10x 2 x 3 Use this model and calculus to answer the following questions. (a) (b) (c) Find dp dx. Hence find the values of x at the stationary points of Find d2 P dx 2. P ¼ 400x 10x 2 x 3 [4 marks] (d) (e) Use your answer to part (c) to find which of the values of x found in part (b) give a maximum value for the profit. Find the maximum profit made in one month by Chloe as predicted by this model. Answer space for question 2 (06)

7 7 Answer space for question 2 Turn over s (07)

8 8 Answer space for question 2 (08)

9 9 Answer space for question 2 Turn over s (09)

10 10 Section C Answer all questions. Answer each question in the space provided for that question. Use Restaurant on page 4 of the Data Sheet. 3 The number of people per hour, n, going into the restaurant can be modelled by n ¼ 16 þ 3t 2 þ 3t 3 t 4 for 0 4 t 4 4 where t is the number of hours after 6pm. The total number of people going into the restaurant can be modelled by ð 4 0 ð16 þ 3t 2 þ 3t 3 t 4 Þ dt (a) (b) (i) Use the trapezium rule with four strips to find an estimate for the total number of people going into the restaurant. [5 marks] Use integration to find the value of ð 4 0 ð16 þ 3t 2 þ 3t 3 t 4 Þ dt [4 marks] (ii) Hence give an estimate of the average number of people going into the restaurant per hour. (c) Would the use of the trapezium rule have changed the estimate of the average number of people going into the restaurant per hour and by how much? Answer space for question 3 (10)

11 11 Answer space for question 3 Turn over s (11)

12 12 Answer space for question 3 (12)

13 13 Answer space for question 3 Turn over s (13)

14 14 Section D Answer all questions. Answer each question in the space provided for that question. Use Footwear manufacturer on page 5 of the Data Sheet. 4 Jack drew up a target for increasing the value of the company s sales by assuming that the value of the company s sales may be modelled by the differential equation ds dt ¼ 0:06S where S is the value of the company s annual sales of footwear and t is the time in years. The time t ¼ 0 corresponds to the year (a) (i) Solve this differential equation to find the general solution for S in terms of t. [3 marks] (ii) Show that S ¼ S 1 e 0:06t where S 1 denotes the value of the annual sales in (b) Find the sales predicted by this model in Express your answer in terms of S 1. (c) In which year does Jack expect the annual sales to have doubled in value from their value in 2012? Answer space for question 4 (14)

15 15 Answer space for question 4 Turn over s (15)

16 16 5 Jack decides that the number of people, y, he intends to employ may be modelled by y ¼ 5 þ 20t 3 e t where t is the time in years. The time t ¼ 0 corresponds to the year (a) Show that dy dt ¼ 20t 2 ð3 tþe t. [4 marks] (b) Hence prove that the number of employees predicted by this model is a maximum when t ¼ 3. (You must show that the model predicts a maximum and not a minimum.) [3 marks] (c) Find the maximum number of employees predicted by this model. Give your answer to the nearest integer. Answer space for question 5 (16)

17 17 Answer space for question 5 Turn over s (17)

18 18 Section E Answer all questions. Answer each question in the space provided for that question. Use Trampoline on page 6 of the Data Sheet. 6 When Alice uses the trampoline, the distance of the top of Alice s head, h centimetres, above the ground can be modelled by h ¼ 250 þ 80 sin p 5 t where t is the number of seconds after Claire starts to watch her. (a) Find the distance of the top of Alice s head above the ground 2.5 seconds after Claire starts to watch her. [1 mark] (b) Find an expression for dh dt. (c) Verify that the model predicts that the distance of the top of Alice s head above the ground has a minimum value 7.5 seconds after Claire starts to watch her. Answer space for question 6 (18)

19 19 Answer space for question 6 Turn over s (19)

20 20 Answer space for question 6 Copyright ª 2015 AQA and its licensors. All rights reserved. END OF S (20)

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