# DIFFERENTIATION OPTIMIZATION PROBLEMS

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1 DIFFERENTIATION OPTIMIZATION PROBLEMS

2 Question 1 (***) 4cm 64cm figure 1 figure An open bo is to be made out of a rectangular piece of card measuring 64 cm by 4 cm. Figure 1 shows how a square of side length cm is to be cut out of each corner so that the bo can be made by folding, as shown in figure. a) Show that the volume of the bo, V cm, is given by V = b) Show further that the stationary points of V occur when = 0. c) Find the value of for which V is stationary. (You may find the fact 4 16 = 84 useful.) d) Find, to the nearest cm, the maimum value for V, justifying that it is indeed the maimum value. = 16, V ma 79

3 Question (***) The figure below shows the design of a fruit juice carton with capacity of 1000 cm. The design of the carton is that of a closed cuboid whose base measures cm by cm, and its height is h cm. h a) Show that the surface area of the carton, A 000 A = 4 +. b) Find the value of for which A is stationary. cm, is given by c) Calculate the minimum value for A, justifying fully the fact that it is indeed the minimum value of A. = , Amin 64

4 Question (***) The figure below shows a solid brick, in the shape of a cuboid, measuring 5 cm by cm by h cm. h 5 The total surface area of the brick is 70 cm. a) Show that the volume of the brick, V cm, is given by 5 V = b) Find the value of for which V is stationary. c) Calculate the maimum value for V, fully justifying the fact that it is indeed the maimum value for V. = , V ma =

5 Question 4 (***) h 4 The figure above shows a bo in the shape of a cuboid with a rectangular base cm by 4 cm and no top. The height of the bo is h cm. It is given that the surface area of the bo is 178 cm. a) Show clearly that 864 h =. 5 b) Use part (a) to show that the volume of the bo, V ( ) V = c) Find the value of for which V is stationary. cm, is given by d) Find the maimum value for V, fully justifying the fact that it is the maimum. = 1, V ma = 559.6

6 Question 5 (***) The figure below shows the design of a large water tank in the shape of a cuboid with a square base and no top. The square base is of length metres and its height h metres. h It is given that the volume of the tank is 500 m. a) Show that the surface area of the tank, A m, is given by 000 A = +. b) Find the value of for which A is stationary. c) Find the minimum value for A, fully justifying the fact that it is the minimum. = 10, A min = 00

7 Question 6 (***) The figure below shows a pentagon ABCDE whose measurements, in cm, are given in terms of and y. A 8 6 E B y D 10 C a) If the perimeter of the pentagon is 10 cm, show clearly that its area, A cm, is given by A = b) Find, by differentiation, the value of for which A is stationary. c) Calculate the maimum value for A, fully justifying the fact that it is indeed the maimum value of A. = 5, A 8 ma = 97.5

8 Question 7 (***) G E F c 1 A y y D B C The figure above shows a clothes design consisting of two identical rectangles attached to each of the straight sides of a circular sector of radius cm. The rectangles measure cm by y cm and the circular sector subtends an angle of one radian at the centre. The perimeter of the design is 40 cm. a) Show that the area of the design, A cm, is given by A = 0. b) Determine by differentiation the value of for which A is stationary. c) Show that the value of found in part (b) gives the maimum value for A. d) Find the maimum area of the design. = 10, A ma = 100

9 Question 8 (***+) r h The figure above shows a closed cylindrical can, of radius r cm and height h cm. a) Given that the total surface area of the can is 19π cm, show that the volume of the can, V cm, is given by V = 96π r π r. b) Find the value of r for which V is stationary. c) Justify that the value of r found in part (b) gives the maimum value for V. d) Calculate the maimum value of V. r = , Vma = 56π 117

10 Question 9 (***+) r h A pencil holder is in the shape of a right circular cylinder, which is open at one of its circular ends. The cylinder has radius r cm and height h cm and the total surface area of the cylinder, including its base, is 60 cm. a) Show that the volume, V cm, of the cylinder is given by 1 V = 180r π r. b) Determine by differentiation the value of r for which V has a stationary value. c) Show that the value of r found in part (b) gives the maimum value for V. d) Calculate, to the nearest cm, the maimum volume of the pencil holder. 10 r = 6.18, Vma 74 π

11 Question 10 (***+) 5 15 y 0 The figure above shows a solid triangular prism with a total surface area of 600 cm. The triangular faces of the prism are right angled with a base of 0 cm and a height of 15 cm. The length of the prism is y cm. a) Show that the volume of the prism, V cm, is given by V = b) Find the value of for which V is stationary. c) Show that the value of found in part (b) gives the maimum value for V. d) Determine the value of y when V becomes maimum. =, y = 0

12 Question 11 (***+) r h The figure above shows a closed cylindrical can, of radius r cm and height h cm. a) If the volume of the can is 0 given by cm, show that surface area of the can, A 660 A = π r +. r b) Find the value of r for which A is stationary. c) Justify that the value of r found in part (b) gives the minimum value for A. d) Hence calculate the minimum value of A. cm, is r.745, Amin 64

13 Question 1 (***+) h ( l ) l The figure above shows 1 rigid rods, joined together to form the framework of a storage container, which in the shape of a cuboid. Each of the four upright rods has height h m. Each of the longer horizontal rods has l m. length l m and each of the shorter horizontal rods have length ( ) a) Given that the total length of the 1 rods is 6 m show that the volume, V of the container satisfies m, V = l + 15l l. b) Find, correct to decimal places, the value of l which make V stationary. c) Justify that the value of l found in part (b) maimizes the value of V, and find this maimum value of V, correct to the nearest m. d) State the three measurements of the container when its volume is maimum. l = 4.107, Vma 4,

14 Question 1 (***+) r h A container is in the shape of a right circular cylinder, which is open at one of its circular ends. The cylinder has radius r cm and height h cm. The volume of the cylinder is 1500 a) Show that the surface area, A cm. cm, of the cylinder is given by 000 A = π r +. r b) Determine the value of r for which A has a stationary value. c) Show that the value of r found in part (b) gives the minimum value for A. d) Calculate, to the nearest cm, the minimum surface area of the container. r 7.816, Amin 576

15 Question 14 (***+) A circular sector of radius cm subtends an angle of θ radians at the centre. θ The area of the sector is 6 a) Show clearly that cm and its perimeter is P cm. 7 P = +. b) Use differentiation to find the value of for which P is stationary. c) Find the minimum value of P, fully justifying the fact that it is a minimum. d) Determine the value of θ when P is minimum. c = 6, P min = 4, θ =

16 Question 15 (***+) The figure below shows a large tank in the shape of a cuboid with a rectangular base and no top. Two of the vertical opposite faces of the cuboid are square and the height of the cuboid is metres. a) Given that the surface area of the tank is 54 tank, V m, is given by m, show that the capacity of the V = 18. b) Find the maimum value for V, fully justifying the fact that it is indeed the maimum value. V ma = 6

17 Question 16 (***+) θ A wire of total length 60 cm is to be cut into two pieces. The first and piece is bent to form an equilateral triangle of side length cm and the second piece is bent to form a circular sector of radius cm. The circular sector subtends an angle of θ radians at the centre. a) Show that The total area of the two shapes is A b) Show clearly that θ = cm. 1 ( ) A = c) Use differentiation to find the value of for which A is stationary. d) Find, to three significant figures, the maimum value of A, justifying the fact that it is the maimum value of A. 7.6, Ama 109

18 Question 17 (****) y The figure above shows the design of a theatre stage which is the shape of a semicircle attached to rectangle. The diameter of the semicircle is m and is attached to one side of the rectangle also measuring m. The other side of the rectangle is y m. The perimeter of the stage is 60 m. a) Show that the total area of the stage, A m, is given by 1 A = 60 π. b) Determine by differentiation an eact value of for which A is stationary. c) Show that the value of found in part (b) gives the maimum value for A. d) Show clearly that the maimum area of the stage is 1800 π + 4 m. 60 = 8.40 π + 4

19 Question 18 (****) r The figure above shows the design of an athletics track inside a stadium. The track consists of two semicircles, each of radius r m, joined up to a rectangular section of length metres. The total length of the track is 400 m and encloses an area of A m. a) By obtaining and manipulating epressions for the total length of the track and the area enclosed by the track, show that A = 400r π r. In order to hold field events safely, it is required for the area enclosed by the track to be as large as possible. b) Determine by differentiation an eact value of r for which A is stationary. c) Show that the value of r found in part (b) gives the maimum value for A. d) Show further that the maimum area enclosed by the track is π m. [continues overleaf]

20 [continued from overleaf] The calculations for maimizing the area of the field within the track are shown to a mathematician. The mathematician agrees that the calculations are correct but he feels the resulting shape of the track might not be suitable. e) Eplain, by calculations, the mathematician s reasoning. 00 r = 6.66 π

21 Question 19 (****) y y The figure above shows the design for an earring consisting of a quarter circle with two identical rectangles attached to either straight edge of the quarter circle. The quarter circle has radius cm and the each of the rectangles measure cm by y cm. The earring is assumed to have negligible thickness and treated as a two dimensional object with area 1.5 cm. a) Show that the perimeter, P cm, of the earring is given by 49 P = +. b) Find the value of that makes the perimeter of the earring minimum, fully justifying that this value of produces a minimum perimeter. c) Show that for the value of found in part (b), the corresponding value of y is ( π ) =.5

22 Question 0 (****) The figure below shows the design of an animal feeder which in the shape of a hollow, open topped half cylinder, made of thin sheet metal. The radius of the semicircular ends is r cm and the length of the feeder is L cm. The metal used in the construction of the feeder is 600π cm. L r a) Show that the capacity, V cm, of the feeder is given by 1 V = 00π r π r. The design of the feeder is such so its capacity is maimum. b) Determine the eact value of r for which V is stationary. c) Show that the value of r found in part (b) gives the maimum value for V. d) Find, in eact form, the capacity and the length of the feeder. r = , L = 0 8.8, Vma = 000π 8886

23 Question 1 (****) y The figure above shows the design of a window which is the shape of a semicircle attached to rectangle. The diameter of the semicircle is m and is attached to one side of the rectangle also measuring m. The other side of the rectangle is y m. The perimeter of the window is 6 m. a) Show that the total area of the window, A m, is given by 1 ( ) A = π. b) Given that the measurements of the window are such so that A is maimum, show clearly that this maimum value of A is π proof

24 Question (****) The figure below shows the design of a hazard warning logo which consists of three identical sectors of radius r cm, joined together at the centre. Each sector subtends an angle θ radians at the centre and the sectors are equally spaced so that the logo has rotational symmetry of order. r θ The area of the logo is 75 cm. a) Show that the perimeter P cm of the logo is given by 150 P = 6r +. r b) Determine by differentiation the value of r for which P is stationary. c) Show that the value of r found in part (b) gives the minimum value for P. d) Find the minimum perimeter of the feeder. r = 5, P min = 60

25 Question (****) 6 L 8 The figure above shows a triangular prism with a volume of 960 cm. The triangular faces of the prism are right angled with a base 8 cm and a height of 6 cm. The length of the prism is L. a) Show that the surface area of the prism, A 960 A = cm, is given by b) Determine by differentiation an eact value of for which A is stationary. c) Show that the value of found in part (b) gives the minimum value for A. d) Show that the minimum surface area of the prism is cm =

26 Question 4 (****) A 4r D O θ r R C B The figure above shows a circular sector OAB of radius 4r subtending an angle θ radians at the centre O. Another circular sector OCD of radius r also subtending an angle θ radians at the centre O is removed from the first sector leaving the shaded region R. It is given that R has an area of 50 square units. a) Show that the perimeter P, of the region R, is given by 100 P = r +. r b) Given that the value of r can vary, i. find an eact value of r for which P is stationary. ii. show that the value of r found above gives the minimum value for P. c) Calculate the minimum value of P. r = , P min = 0 8.8

27 Question 5 (****) y The figure above shows a triangular prism whose triangular faces are parallel to each other and are in the shape of equilateral triangles of side length cm. The length of the prism is y. a) Given that total surface area of the prism is eactly 54 the volume of the prism, V cm, is given by cm, show clearly that 7 1 V =. 8 b) Find the maimum value of V, fully justifying the fact that it is indeed the maimum value. c) Determine the value of y when V takes this maimum value. V ma = 7, y =

28 Question 6 (****) h c r The figure above shows solid right prism of height h cm. The cross section of the prism is a circular sector of radius r cm, subtending an angle of radians at the centre. a) Given that the volume of the prism is 1000 where S 4000 S = r +, r cm is the total surface area of the prism. cm, show clearly that b) Hence determine the value of r and the value of h which make S least, fully justifying your answer. r = 10, h = 10

29 Question 7 (****) A tank is in the shape of a closed right circular cylinder of radius r m and height h m. The tank has a volume of 16π m and is made of thin sheet metal. Given the surface area of the tank is a minimum, determine the value of r and the value of h. r =, h = 4

30 Question 8 (****+) H 1 G D C E y F A 10 B The figure above shows the design of a baking tray with a horizontal rectangular base ABCD, measuring 10 cm by y cm. The faces ABFE and DCGH are isosceles trapeziums, parallel to each other. The lengths of the edges EF and HG are 1 cm. The faces ADHE and BCGF are identical rectangles. The height of the tray is cm. The capacity of the tray is 1980 cm. a) Show that the surface area, A cm, of the tray is given by 60 ( ) A = b) Determine by differentiation the value of for which A is stationary. [continues overleaf]

31 [continued from overleaf] c) Show that the value of found in part (b) gives the minimum value for A. d) Show that the minimum surface area of the tray is approimately 74 cm..744

32 Question 9 (****+) A C D F B E The figure above shows the design of a horse feeder which in the shape of a hollow, open topped triangular prism. The triangular faces at the two ends of the feeder are isosceles and right angled, so that AB = BC = DE = EF and ABC = DEF = 90. The triangular faces are vertical, and the edges AD, BE and CF are horizontal. The capacity of the feeder is 4 m. a) Show that the surface area, A m, of the feeder is given by 1 16 A = +. b) Determine by differentiation the value of for which A is stationary, giving the answer in the form k, where k is an integer. c) Show that the value of found in part (b) gives the minimum value for A. [continues overleaf]

33 [continued from overleaf] d) Show, by eact calculations, that the minimum surface area of the feeder is 1 m. e) Show further that the length ED is equal to the length EB. =.8, ED = EB =

34 Question 0 (****+) The figure below shows the design of a window which is the shape of a semicircle attached to rectangle. y The diameter of the semicircle is metres and is attached to one side of the rectangle also measuring meters. The other side of the rectangle is y metres. The total area of the window is m. a) Show that perimeter, P m, is given by P = 1 ( 4 + π ) +. b) Determine by differentiation an eact value of for which P is stationary. [continues overleaf]

35 [continued from overleaf] c) Show that the value of found in part (b) gives the minimum value for P. d) Show that when P takes a minimum value = y. = π + 4

36 Question 1 (****+) r h r The figure above shows a hollow container consisting of a right circular cylinder of radius r cm and of height h cm joined to a hemisphere of radius r cm. The cylinder is open on one of the circular ends and the hemisphere is also open on its circular base. The cylinder is joined to the hemisphere at their open ends so that the resulting object is completely sealed. a) Given that volume of the container is eactly 880π cm, show clearly that the total surface area of the container, S cm, is given by ( r 456) 5π S = +. r b) Show further than when S is minimum, r = h. proof

37 Question (****+) cm cm cm The figure above shows the design of coffee jar with a push on lid. The jar is in the shape of a right circular cylinder of radius cm. It is fitted with a lid of width cm, which fits tightly on the top of the jar, so it may be assumed that it has the same radius as the jar. The jar and its lid is made of sheet metal and there is no wastage. The total metal used to make the jar and its lid is 190π cm. (This figure does not include the handle of the lid which is made of different material.) a) Show that volume of the jar, V cm, is given by ( 95 ) V = π. b) Determine by differentiation the value of for which V is stationary. [continues overleaf]

38 [continued from overleaf] c) Show that the value of found in part (b) gives the maimum value for V. d) Hence determine the maimum volume of the jar. = 5 cm, Vma = 00π 94 cm

39 Question (****+) The profit of a small business, P is modelled by the equation ( y y 4) P =, where and y are positive variables associated with the running of the company. It is further known that and y constrained by the relation + y = 54. a) Show clearly that P = b) Hence show that the stationary value of P produces a maimum value of 96. The owner is very concerned about the very small profit and shows the calculations to a mathematician. The mathematician agrees that the calculations are correct but he asserts that the profit is substantially higher. c) Eplain, by calculations, the mathematician s reasoning. proof

40 Question 4 (*****) L y Q( 8,5) P d O R The straight line L has equation + y = 8. The point P(, y ) lies on L and the point ( 8,5) Q lies outside L. The point R lies on L so that QR is perpendicular to L. The length PQ is denoted by d. a) Show clearly that f = Let ( ) d = b) Use differentiation to find the stationary value of f ( ), fully justifying that this value of minimizes the value of f ( ). c) State the coordinates of R and find, as an eact surd, the shortest distance of the point Q from L =, R (,1), 5

41 Question 5 (*****) An open bo is to be made of thin sheet metal, in the shape of a cuboid with a square base of length and height h. The bo is to have a fied volume. Show that h is half as large as, when the surface area of the bo is minimum. proof Question 6 (*****) A solid right circular cylinder of fied volume has radius r and height h. Show clearly that when the surface area of the cylinder is minimum h : r = :1. proof

42 Question 7 (*****) A solid right circular cylinder is to be cut out of a solid right circular cone, whose radius is 1.5 m and its height is m. The ais of symmetry of the cone coincides with the ais of symmetry of the cylinder which passes though its circular ends. The circumference of one end of the cylinder is in contact with the curved surface of the cone and the other end of the cylinder lies on the base of the cone. Show that the maimum volume of the cylinder to be cut out is π m. CS, proof

43 Question 8 (*****) The point P lies on the curve with equation A ( 10,) is least. y =, so that its distance from the point Determine the coordinates of P and the distance AP. P (,4), d min = 17

44 Question 9 (*****) A Q R B P S C The figure above shows an isosceles triangle ABC, where AB PQRS drawn inside the triangle. = AC and a rectangle The points P and S lie on BC, the point Q lies on AB and the point R lies on AC. Given that the base of the triangle BC is equal in length to its height, show clearly that the largest area that the rectangle PQRS can achieve is 1 the area of the triangle ABC. proof

45 Question 40 (*****) A right circular cone of radius r and height h is to be cut out of a sphere of radius R. It is a requirement that the circumference of the base of the cone and its verte lie on the surface of the sphere. Determine, in eact form in terms of R, and with full justification, the maimum volume of the cone that can be cut out of this sphere. SP-X, V ma π R = 81

46 Question 41 (*****) A mobile phone wholesaler buys a certain brand of phone for 5 a unit and sells it to shops for 100 a unit. In a typical week the wholesaler epects to sell 500 of these phones. Research however showed that on a typical week for every 1 reduced of the selling price of this phone, an etra 0 sales can be achieved. Determine the selling price for this phone if the weekly profit is to be maimized, and find this maimum weekly profit. CT, 80, maimum profit 40500

47 Question 4 (*****) R H R The figure above shows a hollow container consisting of a right circular cylinder of radius R and of height H joined to a hemisphere of radius R. The cylinder is open on one of the circular ends and the hemisphere is also open on its circular base. The cylinder is joined to the hemisphere at their open ends so that the resulting object is completely sealed. Given that volume of the container is V, show the surface area of the container is minimised when R = H, and hence show further that this minimum surface area is 5π V. proof

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