Optimization with constraints

9. Figure 3
\includegraphics[max width=\textwidth, alt={}, center]{de11ae90-8693-4346-b195-1d01f2b164f5-03_670_782_258_1820} Figure 3 shows the plan of a stage in the shape of a rectangle joined to a semicircle. The length of the rectangular part is \(2 x\) metres and the width is \(y\) metres. The diameter of the semicircular part is \(2 x\) metres. The perimeter of the stage is 80 m .

1. Show that the area, \(A \mathrm {~m} ^ { 2 }\), of the stage is given by $$A = 80 x - \left( 2 + \frac { \pi } { 2 } \right) x ^ { 2 }$$
2. Use calculus to find the value of \(x\) at which \(A\) has a stationary value.
3. Prove that the value of \(x\) you found in part (b) gives the maximum value of \(A\).
4. Calculate, to the nearest \(\mathrm { m } ^ { 2 }\), the maximum area of the stage. \section*{6664/01} \section*{Monday 20 June 2005 - Morning} Mathematical Formulae (Green) Items included with question papers Nil Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G. Write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Core Mathematics C2), the paper reference (6664), your surname, initials and signature. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
   Full marks may be obtained for answers to ALL questions.
   There are 10 questions in this question paper. The total mark for this paper is 75. You must ensure that your answers to parts of questions are clearly labelled.
   You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.
   1. Find the coordinates of the stationary point on the curve with equation \(y = 2 x ^ { 2 } - 12 x\).
   \section*{2. Solve}
5. \(5 ^ { x } = 8\), giving your answer to 3 significant figures,
6. \(\log _ { 2 } ( x + 1 ) - \log _ { 2 } x = \log _ { 2 } 7\).
   3. (a) Use the factor theorem to show that \(( x + 4 )\) is a factor of \(2 x ^ { 3 } + x ^ { 2 } - 25 x + 12\).
7. Factorise \(2 x ^ { 3 } + x ^ { 2 } - 25 x + 12\) completely.
   4. (a) Write down the first three terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 + p x ) ^ { 12 }\), where \(p\) is a non-zero constant. Given that, in the expansion of \(( 1 + p x ) ^ { 12 }\), the coefficient of \(x\) is \(( - q )\) and the coefficient of \(x ^ { 2 }\) is \(11 q\),
8. find the value of \(p\) and the value of \(q\).
   5. Solve, for \(0 \leq x \leq 180 ^ { \circ }\), the equation
9. \(\sin \left( x + 10 ^ { \circ } \right) = \frac { \sqrt { } 3 } { 2 }\),
10. \(\cos 2 x = - 0.9\), giving your answers to 1 decimal place.
    6. A river, running between parallel banks, is 20 m wide. The depth, \(y\) metres, of the river measured at a point \(x\) metres from one bank is given by the formula $$y = \frac { 1 } { 10 } x \sqrt { } ( 20 - x ) , \quad 0 \leq x \leq 20 .$$
11. Complete the table below, giving values of \(y\) to 3 decimal places.
12. Complete the table, giving the values of \(v\) to 2 decimal places. The distance, \(s\) metres, travelled by the train in 30 seconds is given by $$s = \int _ { 0 } ^ { 30 } \sqrt { } \left( 1.2 ^ { t } - 1 \right) \mathrm { d } t$$
13. Use the trapezium rule, with all the values from your table, to estimate the value of \(s\).
    7. The curve \(C\) has equation $$y = 2 x ^ { 3 } - 5 x ^ { 2 } - 4 x + 2 .$$
14. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
15. Using the result from part (a), find the coordinates of the turning points of \(C\).
16. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
17. Hence, or otherwise, determine the nature of the turning points of \(C\).
    8. (a) Find all the values of \(\theta\), to 1 decimal place, in the interval \(0 ^ { \circ } \leq \theta < 360 ^ { \circ }\) for which $$5 \sin \left( \theta + 30 ^ { \circ } \right) = 3 .$$
18. Find all the values of \(\theta\), to 1 decimal place, in the interval \(0 ^ { \circ } \leq \theta < 360 ^ { \circ }\) for which $$\tan ^ { 2 } \theta = 4 .$$ 9. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{de11ae90-8693-4346-b195-1d01f2b164f5-09_410_835_223_310}
    \end{figure} Figure 3 shows the shaded region \(R\) which is bounded by the curve \(y = - 2 x ^ { 2 } + 4 x\) and the line \(y = \frac { 3 } { 2 }\). The points \(A\) and \(B\) are the points of intersection of the line and the curve. Find
19. the \(x\)-coordinates of the points \(A\) and \(B\),
20. the exact area of \(R\). Materials required for examination
    Mathematical Formulae (Green)
    Items included with question papers
    Nil Reference(s)
    6664/01 Core Mathematics C2
    Advanced Subsidiary
    Monday 22 May 2006 - Morning
    Time: 1 hour 30 minutes Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G. Write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Core Mathematics C2), the paper reference (6664), your surname, initials and signature. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
    Full marks may be obtained for answers to ALL questions.
    There are 10 questions in this question paper. The total mark for this paper is 75 . You must ensure that your answers to parts of questions are clearly labelled.
    You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.
    1. Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of \(( 2 + x ) ^ { 6 }\), giving each term in its simplest form.
    2. Use calculus to find the exact value of \(\int _ { 1 } ^ { 2 } \left( 3 x ^ { 2 } + 5 + \frac { 4 } { x ^ { 2 } } \right) \mathrm { d } x\).
    3. 1. Write down the value of \(\log _ { 6 } 36\).
       2. Express \(2 \log _ { a } 3 + \log _ { a } 11\) as a single logarithm to base \(a\).
    $$\mathrm { f } ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 29 x - 60$$
21. Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x + 2\) ).
22. Use the factor theorem to show that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\).
23. Factorise \(\mathrm { f } ( x )\) completely.
    5. (a) Sketch the graph of \(y = 3 ^ { x } , x \in \mathbb { R }\), showing the coordinates of the point at which the graph meets the \(y\)-axis.
24. Copy and complete the table, giving the values of \(3 ^ { x }\) to 3 decimal places.

    \(x\)	0	0.2	0.4	0.6	0.8	1
    \(3 ^ { x }\)		1.246	1.552			3

25. Use the trapezium rule, with all the values from your tables, to find an approximation for the value of \(\int _ { 0 } ^ { 1 } 3 ^ { x } \mathrm {~d} x\).
    6. (a) Given that \(\sin \theta = 5 \cos \theta\), find the value of \(\tan \theta\).
26. Hence, or otherwise, find the values of \(\theta\) in the interval \(0 \leq \theta < 360 ^ { \circ }\) for which \(\sin \theta = 5 \cos \theta\), giving your answers to 1 decimal place.
    7. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{de11ae90-8693-4346-b195-1d01f2b164f5-10_486_615_625_1866}
    \end{figure} The line \(y = 3 x - 4\) is a tangent to the circle \(C\), touching \(C\) at the point \(\mathrm { P } ( 2,2 )\), as shown in Figure 1. The point \(Q\) is the centre of \(C\).
27. Find an equation of the straight line through \(P\) and \(Q\). Given that \(Q\) lies on the line \(y = 1\),
28. show that the \(x\)-coordinate of \(Q\) is 5 ,
29. find an equation for \(C\).
    8. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{de11ae90-8693-4346-b195-1d01f2b164f5-11_440_474_294_548}
    \end{figure} Figure 2 shows the cross-section \(A B C D\) of a small shed.
    The straight line \(A B\) is vertical and has length 2.12 m .
    The straight line \(A D\) is horizontal and has length 1.86 m .
    The curve \(B C\) is an arc of a circle with centre \(A\), and \(C D\) is a straight line.
    Given that the size of \(\angle B A C\) is 0.65 radians, find
30. the length of the arc \(B C\), in m , to 2 decimal places,
31. the area of the sector \(B A C\), in \(\mathrm { m } ^ { 2 }\), to 2 decimal places,
32. the size of \(\angle C A D\), in radians, to 2 decimal places,
33. the area of the cross-section \(A B C D\) of the shed, in \(\mathrm { m } ^ { 2 }\), to 2 decimal places.
    9. A geometric series has first term \(a\) and common ratio \(r\). The second term of the series is 4 and the sum to infinity of the series is 25 .
34. Show that \(25 r ^ { 2 } - 25 r + 4 = 0\).
35. Find the two possible values of \(r\).
36. Find the corresponding two possible values of \(a\).
37. Show that the sum, \(S _ { n }\), of the first \(n\) terms of the series is given by $$S _ { n } = 25 \left( 1 - r ^ { n } \right)$$ Given that \(r\) takes the larger of its two possible values,
38. find the smallest value of \(n\) for which \(S _ { n }\) exceeds 24 .
    \includegraphics[max width=\textwidth, alt={}, center]{de11ae90-8693-4346-b195-1d01f2b164f5-12_442_899_342_310} \section*{Edexcel GCE
    Core Mathematics C2 Advanced Subsidiary} Materials required for examination
    Mathematical Formulae (Green) \section*{Wednesday 10 January 2007 - Afternoon Time: 1 hour 30 minutes} Items included with question papers Nil Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP \(\mathbf { 4 8 G }\). Write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Core Mathematics C2), the paper reference (6664), your surname, initials and signature. A booklet 'Mathematical Formulae and Statistical Tables' is provided. Full marks may be obtained for answers to ALL questions. There are 10 questions in this question paper. The total mark for this paper is 75 . You must ensure that your answers to parts of questions are clearly labelled. You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit. \footnotetext{N24322A
    This publication may only be reproduced in accordance with Edexcel Limited copyright policy. ©2007 Edexcel Limited. } HP 48G. . .
    新新 \section*{TOTAL FOR PAPER: 75 MARKS} \section*{END}
39. Hence calculate the exact area of \(R\). The line through \(B\) parallel to the \(y\)-axis meets the \(x\)-axis at the point \(N\). The region \(R\), shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and the line from \(A\) to \(N\).
40. Find \(\int \left( x ^ { 3 } - 8 x ^ { 2 } + 20 x \right) \mathrm { d } x\).
41. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\), and hence verify that \(A\) is a maximum. \includegraphics[max width=\textwidth, alt={}]{de11ae90-8693-4346-b195-1d01f2b164f5-12_74_49_929_1283} 2
    "
    1. $$\mathrm { f } ( x ) = x ^ { 3 } + 3 x ^ { 2 } + 5 .$$ Find
42. \(\mathrm { f } ^ { \prime \prime } ( x )\),
43. \(\int _ { 1 } ^ { 2 } f ( x ) d x\).
    2. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 - 2 x ) ^ { 5 }\). Give each term in its simplest form.
44. If \(x\) is small, so that \(x ^ { 2 }\) and higher powers can be ignored, show that $$( 1 + x ) ( 1 - 2 x ) ^ { 5 } \approx 1 - 9 x .$$
    1. The line joining points \(( - 1,4 )\) and \(( 3,6 )\) is a diameter of the circle \(C\).
    Find an equation for \(C\).
    4. Solve the equation \(5 ^ { x } = 17\), giving your answer to 3 significant figures.
    5. $$f ( x ) = x ^ { 3 } + 4 x ^ { 2 } + x - 6$$
45. Use the factor theorem to show that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\).
46. Factorise \(\mathrm { f } ( x )\) completely.
47. Write down all the solutions to the equation $$x ^ { 3 } + 4 x ^ { 2 } + x - 6 = 0 .$$
    1. Find all the solutions, in the interval \(0 \leq x < 2 \pi\), of the equation
    $$2 \cos ^ { 2 } x + 1 = 5 \sin x ,$$ giving each solution in terms of \(\pi\).
    7. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{de11ae90-8693-4346-b195-1d01f2b164f5-13_768_826_523_1731}
    \end{figure} Figure 1 shows a sketch of part of the curve \(C\) with equation $$y = x ( x - 1 ) ( x - 5 ) .$$ Use calculus to find the total area of the finite region, shown shaded in Figure 1, that is between \(x = 0\) and \(x = 2\) and is bounded by \(C\), the \(x\)-axis and the line \(x = 2\).
    (9)
    8. A diesel lorry is driven from Birmingham to Bury at a steady speed of \(v\) kilometres per hour. The total cost of the journey, \(\pounds C\), is given by $$C = \frac { 1400 } { v } + \frac { 2 v } { 7 } .$$
48. Find the value of \(v\) for which \(C\) is a minimum.
49. Find \(\frac { \mathrm { d } ^ { 2 } C } { \mathrm {~d} v ^ { 2 } }\) and hence verify that \(C\) is a minimum for this value of \(v\).
50. Calculate the minimum total cost of the journey.
    9. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{de11ae90-8693-4346-b195-1d01f2b164f5-14_433_725_769_397}
    \end{figure} Figure 2 shows a plan of a patio. The patio \(P Q R S\) is in the shape of a sector of a circle with centre \(Q\) and radius 6 m . Given that the length of the straight line \(P R\) is \(6 \sqrt { } 3 \mathrm {~m}\),
51. find the exact size of angle \(P Q R\) in radians.
52. Show that the area of the patio \(P Q R S\) is \(12 \pi \mathrm {~m} ^ { 2 }\).
53. Find the exact area of the triangle \(P Q R\).
54. Find, in \(\mathrm { m } ^ { 2 }\) to 1 decimal place, the area of the segment PRS.
55. Find, in m to 1 decimal place, the perimeter of the patio \(P Q R S\).

5. \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 $$A = 4 x ^ { 2 } + \frac { 3090 } { x } .$$ The manufacturer needs to minimise the surface area of a carton.
3. Use calculus to find the value of \(x\) for which \(A\) is a minimum.
4. Calculate the minimum value of \(A\).
5. Prove that this value of \(A\) is a minimum.

3 Two numbers, \(x\) and \(y\), are such that \(3 x + y = 9\), where \(x \geqslant 0\) and \(y \geqslant 0\). It is given that \(V = x y ^ { 2 }\).

1. Show that \(V = 81 x - 54 x ^ { 2 } + 9 x ^ { 3 }\).
2. 1. Show that \(\frac { \mathrm { d } V } { \mathrm {~d} x } = k \left( x ^ { 2 } - 4 x + 3 \right)\), and state the value of the integer \(k\).
   2. Hence find the two values of \(x\) for which \(\frac { \mathrm { d } V } { \mathrm {~d} x } = 0\).
3. Find \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }\).
4. 1. Find the value of \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }\) for each of the two values of \(x\) found in part (b)(ii).
   2. Hence determine the value of \(x\) for which \(V\) has a maximum value.
   3. Find the maximum value of \(V\).

7 Given that \(y \in \mathbb { R }\), prove that $$( 2 + 3 y ) ^ { 4 } + ( 2 - 3 y ) ^ { 4 } \geq 32$$ Fully justify your answer.
[0pt] [6 marks]

11 Rakti makes open-topped cylindrical planters out of thin sheets of galvanised steel. bends a rectangle of steel to make an open cylinder and welds the joint. She She bends this cylinder to the circumference of a circular base then welds this cylinder to the circumference of a circular base.
\includegraphics[max width=\textwidth, alt={}, center]{8d9ace4b-0c15-48bd-9b0d-302f57ea9759-12_552_524_497_753} The planter must have a capacity of \(8000 \mathrm {~cm} ^ { 3 }\) Welding is time consuming, so Rakti wants the total length of weld to be a minimum.
Calculate the radius, \(r\), and height, \(h\), of a planter which requires the minimum total length of weld. Fully justify your answers, giving them to an appropriate degree of accuracy.

10 A square sheet of metal has edges 30 cm long. Four squares each with edge \(x \mathrm {~cm}\), where \(x < 15\), are removed from the corners of the sheet. The four rectangular sections are bent upwards to form an open-topped box, as shown in the diagrams.
\includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-12_392_460_630_347}
\includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-12_387_437_635_872}
\includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-12_282_380_703_1318} 10

1. Show that the capacity, \(C \mathrm {~cm} ^ { 3 }\), of the box is given by $$C = 900 x - 120 x ^ { 2 } + 4 x ^ { 3 }$$ 10
2. Find the maximum capacity of the box. Fully justify your answer.

10 A piece of wire of length 66 cm is bent to form the five sides of a pentagon. The pentagon consists of three sides of a rectangle and two sides of an equilateral triangle. The sides of the rectangle measure \(x \mathrm {~cm}\) and \(y \mathrm {~cm}\) and the sides of the triangle measure \(x \mathrm {~cm}\), as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{e3635007-2ad1-4b2a-b937-41fe90bb1111-12_405_492_630_863} 10

1. 1. You are given that \(\sin 60 ^ { \circ } = \frac { \sqrt { 3 } } { 2 }\)
      Explain why the area of the triangle is \(\frac { \sqrt { 3 } } { 4 } x ^ { 2 }\) 10
2. (ii) Show that the area enclosed by the wire, \(A \mathrm {~cm} ^ { 2 }\), can be expressed by the formula $$A = 33 x - \frac { 1 } { 4 } ( 6 - \sqrt { 3 } ) x ^ { 2 }$$ 10
3. Use calculus to find the value of \(x\) for which the wire encloses the maximum area. Give your answer in the form \(p + q \sqrt { 3 }\), where \(p\) and \(q\) are integers. Fully justify your answer.
   \(L _ { 1 }\) is a tangent to the circle \(C\) at the point \(P ( 6,5 )\)
   The line \(L _ { 2 }\) has equation \(y = x + 3\)
   \(L _ { 2 }\) is a tangent to the circle \(C\) at the point \(Q ( 0,3 )\)
   The lines \(L _ { 1 }\) and \(L _ { 2 }\) and the circle \(C\) are shown in the diagram below.
   \includegraphics[max width=\textwidth, alt={}, center]{e3635007-2ad1-4b2a-b937-41fe90bb1111-14_702_714_726_753}

13 A company is designing a logo. The logo is a circle of radius 4 inches with an inscribed rectangle. The rectangle must be as large as possible. The company models the logo on an \(x - y\) plane as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{85b10472-8149-4387-999f-2ef153f1a105-20_492_492_511_776} Use calculus to find the maximum area of the rectangle.
Fully justify your answer.

14 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 \(60 \mathrm {~m} ^ { 3 }\)
The materials cost:

• \(\pounds 15\) per \(\mathrm { m } ^ { 2 }\) for the base
• \(\pounds 8\) per \(\mathrm { m } ^ { 2 }\) for the sides.

14

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.
   [0pt] [8 marks] 14
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 of the tank.
      [0pt] [1 mark]
      LIH
      L
      LL 14
3. (ii) How would your refinement affect your answer to part (a)?
   [0pt] [1 mark]

9 A cylinder is to be cut out of the circular face of a solid hemisphere. The cylinder and the hemisphere have the same axis of symmetry.
The cylinder has height \(h\) and the hemisphere has a radius of \(R\).
\includegraphics[max width=\textwidth, alt={}, center]{27339c29-c4a1-480c-b882-930f8dacc7af-16_467_792_534_625} 9

1. Show that the volume, \(V\), of the cylinder is given by $$V = \pi R ^ { 2 } h - \pi h ^ { 3 }$$ 9
2. Find the maximum volume of the cylinder in terms of \(R\). Fully justify your answer.

7 The curve \(y = 15 - x ^ { 2 }\) and the isosceles triangle \(O P Q\) are shown on the diagram The curve \(y = 15 - x ^ { 2 }\) and the isosceles triangle \(O P Q\) are shown on the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-10_759_810_388_614} Vertices \(P\) and \(Q\) lie on the curve such that \(Q\) lies vertically above some point ( \(q , 0\) ) The line \(P Q\) is parallel to the \(x\)-axis. 7

1. Show that the area, \(A\), of the triangle \(O P Q\) is given by $$A = 15 q - q ^ { 3 } \quad \text { for } 0 < q < c$$ where \(c\) is a constant to be found.
   7
2. Find the exact maximum area of triangle \(O P Q\). Fully justify your answer.