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1 solved problems on optimization

2 Mika Seppälä: Solved Problems on Optimization review of the subject

3 Mika Seppälä: Solved Problems on Optimization STEPS FOR SOLVING OPTIMIZATION PROBLEMS 1. Read the problem carefully until you can answer the questions: a. What are the given quantities? b. What are the given conditions? c. What is the unknown?

4 Mika Seppälä: Solved Problems on Optimization 2. Draw a diagram that illustrates the given and required quantities. 3. Define variables to label the quantities in your diagram. 4. Create your equations. One of them will be the equation to optimize. STEPS FOR SOLVING OPTIMIZATION PROBLEMS

5 Mika Seppälä: Solved Problems on Optimization STEPS FOR SOLVING OPTIMIZATION PROBLEMS 5. Express the equation to optimize as a function of one variable using the other equations. 6. Differentiate this equation with one variable. 7. Verify that your result is maximum or minimum.

6 Mika Seppälä: Solved Problems on Optimization FIRST DERIVATIVE TEST Let c be a critical number of f. (a) If for all and for all, then is the absolute maximum of f. (b) If for all and for all, then is the absolute minimum of f.

7 Mika Seppälä: Solved Problems on Optimization overview of problems

8 Mika Seppälä: Solved Problems on Optimization 1 OVERVIEW OF PROBLEMS Find two numbers whose difference is 100 and whose product is minimum. Find two nonnegative numbers whose sum is 9 and so that the product of one number with the square of the other one is maximum. 2

9 Mika Seppälä: Solved Problems on Optimization OVERVIEW OF PROBLEMS Find the dimensions of a rectangle with perimeter 100 m. whose area is as large as possible. Build a rectangular pen with three parallel partitions using 500 ft. of fencing. What dimensions will maximize the total area of the pen? 3 4

10 Mika Seppälä: Solved Problems on Optimization OVERVIEW OF PROBLEMS A box with a square base and open top must have a volume of 32,000. Find the dimensions of the box that minimize the amount of the material used. 5 Find the equation of the line through the point that cuts off the least area from the first quadrant. 6

11 Mika Seppälä: Solved Problems on Optimization OVERVIEW OF PROBLEMS An open rectangular box with square base is to be made from 48 of material. What should be the dimensions of the box so that it has the largest possible volume? 7

12 Mika Seppälä: Solved Problems on Optimization OVERVIEW OF PROBLEMS A rectangular storage container with an open top is to have a volume of 10. The length of its base is twice the width. Material for the base costs $10 per square meter. Material for the side costs $6 per square meter. Find the cost of materials for the cheapest such container. 8

13 Mika Seppälä: Solved Problems on Optimization OVERVIEW OF PROBLEMS Find the point on the line that is closest to the origin. Find the area of the largest rectangle that can be inscribed in the ellipse 9 10

14 Mika Seppälä: Solved Problems on Optimization OVERVIEW OF PROBLEMS A right circular cylinder is inscribed in a cone with height h and base radius r. Find the largest possible volume of such a cylinder. 11

15 Mika Seppälä: Solved Problems on Optimization OVERVIEW OF PROBLEMS A cone shaped paper drinking cup is to be made to hold 27 of water. Find the height and radius of the cup that will use the smallest amount of paper. 12

16 Mika Seppälä: Solved Problems on Optimization OVERVIEW OF PROBLEMS A boat leaves a dock at 1:00 pm and travels due south at a speed of 20. Another boat heading due east at 15 and reaches the same dock at 2:00 pm. At what time were the two boats closest to each other. 13

17 Mika Seppälä: Solved Problems on Optimization OVERVIEW OF PROBLEMS A football team plays in a stadium that holds 80,000 spectators. With ticket prices at $20, the average attendance had been 51,000. When ticket prices were lowered to $15, the average attendance rose to 66,

18 Mika Seppälä: Solved Problems on Optimization OVERVIEW OF PROBLEMS a. Find the demand function, assuming that it is linear. b.How should ticket prices be set to maximize the revenue?

19 Mika Seppälä: Solved Problems on Optimization OVERVIEW OF PROBLEMS Two vertical poles PQ and ST are secured by a rope PRS going from the top of first pole to a point R on the ground between the poles and then to the top of the second pole as in the figure. 15

20 Mika Seppälä: Solved Problems on Optimization OVERVIEW OF PROBLEMS Show that the shortest length of such a rope occurs when.

21 Mika Seppälä: Solved Problems on Optimization solutions to problems

22 Mika Seppälä: Solved Problems on Optimization optimization Problem 1 Solution Find two numbers whose difference is 100 and whose product is minimum. Let x and y be two numbers such that.

23 Mika Seppälä: Solved Problems on Optimization optimization Solution(contd) The equation we want to minimize is. We can express it as function of x by substituting y using the relation. We derive f:

24 Mika Seppälä: Solved Problems on Optimization optimization Solution(contd) We find that for. We also remark that for,.Then by the first derivative test, we deduce that minimizes f. Hence, the product of these two numbers is minimum when and.

25 Mika Seppälä: Solved Problems on Optimization Problem 2 OPTIMIZATION Find two nonnegative numbers whose sum is 9 and so that the product of one number with the square of the other one is maximum.

26 Mika Seppälä: Solved Problems on Optimization Solution OPTIMIZATION Let x and y be two positive numbers so that. The equation we want to minimize is. We can express it as function of x by substituting y using the relation.

27 Mika Seppälä: Solved Problems on Optimization optimization Solution(contd) We derive f using the product rule: We find that for and.

28 Mika Seppälä: Solved Problems on Optimization For,. Then by the first derivative test, we deduce that maximizes f. optimization Solution(contd) For,. Then by the first derivative test, we deduce that minimizes f.

29 Mika Seppälä: Solved Problems on Optimization optimization Solution(contd) Hence, one of these positive numbers is and the other one is obtained by using the equation.

30 Mika Seppälä: Solved Problems on Optimization Problem 3 OPTIMIZATION Find the dimensions of a rectangle with perimeter 100 m whose area is as large as possible.

31 Mika Seppälä: Solved Problems on Optimization OPTIMIZATION Solution Let x and y be the dimensions of a rectangle x y so that.

32 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION We want to find x and y so that the rectangle has the smallest possible area. That is we want to minimize the equation. We can express the area as function of x by substituting y using the relation.

33 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION We derive f: We find that for. We also remark that for,. Then by the first derivative test, we deduce that maximizes f.

34 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION Hence, one of the dimensions is and the other dimension is obtained by using the equation.

35 Mika Seppälä: Solved Problems on Optimization OPTIMIZATION Problem 4 Build a rectangular pen with three parallel partitions using 500 feet of fencing. What dimensions will maximize the total area of the pen?

36 Mika Seppälä: Solved Problems on Optimization Solution OPTIMIZATION Let x be the length of the pen and y the width of the pen. x y yy y

37 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION The total amount of fencing is given by We want to find x and y that maximize the area of the pen, that is the equation. We can express it as function of x by substituting y using the relation.

38 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION So the area function is We derive f:

39 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION We find that for. We also remark that for,. Then by the first derivative test, we deduce that maximizes f. Using the equation we find the other dimension

40 Mika Seppälä: Solved Problems on Optimization Problem 5 OPTIMIZATION A box with a square base and open top must have a volume of 32,000. Find the dimensions of the box that minimize the amount of the material used.

41 Mika Seppälä: Solved Problems on Optimization Solution OPTIMIZATION z x y Consider the box with dimensions x, y, and z.

42 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION The base of the box is a square, so. The volume of the box is 32,000, so Minimizing the amount of the material used is same as minimizing the sum of the areas of the faces of the box.

43 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION The sum of the areas of the left and the right faces is. The sum of the areas of the front and the back faces is.

44 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION The box does not have top face, so there is only one more face, the bottom face and its area is. The sum of areas is. Next, we express this sum as function of x.

45 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION Since and, we obtain We derive f.

46 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION We find that for. We also remark that for,. Then by the first derivative test, we deduce that minimizes f.

47 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION Using the equations and we find that and.

48 Mika Seppälä: Solved Problems on Optimization Problem 6 OPTIMIZATION Find the equation of the line through the point that cuts off the least area from the first quadrant.

49 Mika Seppälä: Solved Problems on Optimization Solution OPTIMIZATION As shown in the figure, the line L passes through the point and has a as the y- intercept and b as the x- intercept. Then the equation of this line is. line L

50 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION Since the point is on the line L, we have the relation from which we can express b in terms of a.

51 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION The quantity we want to minimize is the area of the triangle aOb which is. We express the area as function of a

52 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION Next, using the quotient rule derive. Then for and.

53 Mika Seppälä: Solved Problems on Optimization Since, by the first derivative test minimizes the area. Solution(contd) OPTIMIZATION We remark that a cannot be 0 because L cuts off an area in the first quadrant. Therefore is the only possible critical value.

54 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION Using the relation between a and b the value of b that minimizes f is 6. Hence, the equation of L that cuts off the least area in the first quadrant is.

55 Mika Seppälä: Solved Problems on Optimization Problem 7 OPTIMIZATION An open rectangular box with square base is to be made from 48 of material. What should be the dimensions of the box so that it has the largest possible volume?

56 Mika Seppälä: Solved Problems on Optimization Solution OPTIMIZATION z x y Consider the box with dimensions x, y, and z.

57 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION The base of the box is a square, so. The amount of the material used to build the box is equal to the sum of the areas of the faces of the box. As in the previous problem this sum is equal to

58 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION We want to find x, y, and z that maximizes the volume of the box, that is we want to maximize the equation. Next, we express this volume as function of x.

59 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION Since and, we obtain We derive f.

60 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION We find that for or. We observe that x cannot be a negative number since it measures a distance. Therefore the only critical point is.

61 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION Since for, the first derivative test implies that maximizes f. Hence, the dimensions of the box with the maximum volume are.

62 Mika Seppälä: Solved Problems on Optimization Problem 8 OPTIMIZATION A rectangular storage container with an open top is to have a volume of 10. The length of its base is twice the width. Material for the base costs $10 per square meter. Material for the side costs $6 per square meter. Find the cost of materials for the cheapest such container.

63 Mika Seppälä: Solved Problems on Optimization Solution OPTIMIZATION z x y Consider the box with dimensions x, y, and z. whose volume is.

64 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION Since the length of the base is twice the width, we have the relation. Then the equation of the volume becomes Hence,.

65 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION Our goal is to minimize the cost function. According to the information given in the problem

66 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION Area of the base is. Total area of the sides is. Therefore the cost function is

67 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION To express the cost function in terms of x, do the substitutions and. Then we obtain,

68 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION Next, we derive the cost function. We find that for. Since for, minimizes the cost function.

69 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION Hence, the minimum cost is

70 Mika Seppälä: Solved Problems on Optimization Find the point on the line that is closest to the origin. Problem 9 OPTIMIZATION Solution Let be the point on the line. Then it satisfies.

71 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION We want to minimize the distance between the point and which is given by the formula

72 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION Using the equation, we express the distance as function of the variable a.

73 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION Using the chain rule, we derive. whenever. Therefore, the critical point is. By the first derivative rule this value of a minimizes the distance.

74 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION This means the x coordinate of the closest point to the origin on the line is. We find the y coordinate using the equation of the line

75 Mika Seppälä: Solved Problems on Optimization Problem 10 OPTIMIZATION Find the area of the largest rectangle that can be inscribed in the ellipse.

76 Mika Seppälä: Solved Problems on Optimization The ellipse is centered at and has vertices at. We assume that a and b are positive. OPTIMIZATION Solution

77 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION We inscribe a rectangle in the ellipse. It is centered at the origin with dimensions 2w and 2h where and.

78 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION We want to find w and h so that the area of the rectangle is largest possible. In other words we want to maximize the area of the rectangle

79 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION We remark that the point is on the ellipse. Therefore from which it follows

80 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION Using the above relation between h and w, we can express the area as function of w. Next, we derive using product and chain rules.

81 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION

82 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION when. Since w and a are positive,. implies from the first derivative test that maximizes the area of the rectangle.

83 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION We compute the value of h from the relation Hence, the area of the largest rectangle that can be inscribed in is

84 Mika Seppälä: Solved Problems on Optimization Problem 11 OPTIMIZATION A right circular cylinder is inscribed in a cone with height h and base radius r. Find the largest possible volume of such a cylinder.

85 Mika Seppälä: Solved Problems on Optimization Solution OPTIMIZATION h r Consider the cone with radius r and height h.

86 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION r h As shown in the figure, we inscribe a right cylinder with radius r and height h in that cone. We want to find r and h so that the volume of the cylinder is largest.

87 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION In other words, we want to maximize the function First, we need to rewrite the Volume function as function of one variable.

88 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION r h r h-h Using the fact that the triangles AOB and AOB are similar, we deduce the relation or

89 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION We rewrite the Volume as function of h.

90 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION Using the chain and product rule, derive.

91 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION when and. If h was equal to h then the cylinder would be out of the cone. Therefore. We compute the derivative at.

92 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION Since, the first derivative test that maximizes the volume. Then the radius of the cylinder is. Hence the maximum volume is

93 Mika Seppälä: Solved Problems on Optimization Problem 12 OPTIMIZATION A cone shaped paper drinking cup is to be made to hold 27 of water. Find the height and radius of the cup that will use the smallest amount of paper.

94 Mika Seppälä: Solved Problems on Optimization Solution OPTIMIZATION r h Consider a cone shaped paper cup whose height is h and radius is r. Since it holds 27 of water, its volume is

95 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION Minimizing the amount of the paper means minimizing the lateral area of the cone. To find the lateral area, we cut the cone open and obtain a triangle.

96 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION The base of this triangle is, the circumference of the base of the cone and the height is, the height of the cone.

97 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION The area of the triangle is We rewrite it as function of r by substituting h

98 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION We derive.

99 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION After simplification. Therefore, when. That is when. We want to point out that the negative value of r is ignored since r measures radius.

100 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION By the first derivative test minimizes the area. For this r we find the height of the cone to be

101 Mika Seppälä: Solved Problems on Optimization Problem 13 OPTIMIZATION A boat leaves a dock at 1:00 pm and travels due south at a speed of 20. Another boat heading due east at 15 and reaches the same dock at 2:00 pm. At what time were the two boats closest to each other.

102 Mika Seppälä: Solved Problems on Optimization Solution OPTIMIZATION Position of the boats at 1:00 pm We call A the boat that travels south and B the boat that travels east. Figure 1 and 2 show their positions at 1:00 pm and 2:00 pm, respectively. Position of the boats at 2:00 pm

103 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION In t hours, B covers 15t km. to the east and A covers 20t km. to the south. The figure shows the positions of the boats with respect to the dock t hours after 1:00 pm.

104 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION t hours after 1:00 pm, the distance between A and B is We want to find t so that is smallest. For that we derive and find its critical points.

105 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION

106 Mika Seppälä: Solved Problems on Optimization Solution(contd) OPTIMIZATION Then whenever or when. Since for, by the first derivative test minimizes. hours is equivalent to 21 minutes and 36 seconds. Hence the boats are closet to each other at 1:21:36.

107 Mika Seppälä: Solved Problems on Optimization OPTIMIZATION Problem 14 A football team plays in a stadium that holds 80,000 spectators. With ticket prices at $20, the average attendance had been 51,000. When ticket prices were lowered to $15, the average attendance rose to 66,000.

108 Mika Seppälä: Solved Problems on Optimization a. Find the demand function, assuming that it is linear. OPTIMIZATION Solution Demand function is a relation between the price and the quantity where price is places on the y-axis and quantity on the x-axis.

109 Mika Seppälä: Solved Problems on Optimization OPTIMIZATION Solution(contd) OPTIMIZATION We are given that the demand function is linear. Therefore it is of the form where x is the average attendance (quantity in economical terms) and is the price.

110 Mika Seppälä: Solved Problems on Optimization OPTIMIZATION Solution(contd) OPTIMIZATION From the question, we understand that the points and are on the line. So the equation of the line passing through these points is

111 Mika Seppälä: Solved Problems on Optimization Solution OPTIMIZATION (b) How should ticket prices be set to maximize the revenue? In economics, revenue function is equal to:

112 Mika Seppälä: Solved Problems on Optimization OPTIMIZATION Solution(contd) OPTIMIZATION In this question, revenue function is

113 Mika Seppälä: Solved Problems on Optimization To maximize, we find its critical points. OPTIMIZATION Solution(contd) OPTIMIZATION for. Since for by the first derivative test maximizes the revenue.

114 Mika Seppälä: Solved Problems on Optimization OPTIMIZATION Solution(contd) OPTIMIZATION Hence, the average attendance that maximizes the revenue is From this, the price that maximizes the revenue is

115 Mika Seppälä: Solved Problems on Optimization OPTIMIZATION Problem 15 Two vertical poles PQ and ST are secured by a rope PRS going from the top of first pole to a point R on the ground between the poles and then to the top of the second pole as in the figure.

116 Mika Seppälä: Solved Problems on Optimization OPTIMIZATION Show that the shortest length of such a rope occurs when.

117 Mika Seppälä: Solved Problems on Optimization Solution OPTIMIZATION Let the height of the poles be and and the distance between the poles be d. We note that these are constant quantities.

118 Mika Seppälä: Solved Problems on Optimization OPTIMIZATION Solution(contd) OPTIMIZATION We also let

119 Mika Seppälä: Solved Problems on Optimization OPTIMIZATION Solution(contd) OPTIMIZATION The length of the rope is where by using the Pythagorean Theorem and. Hence, the length can be expressed as function of x

120 Mika Seppälä: Solved Problems on Optimization OPTIMIZATION Solution(contd) OPTIMIZATION We derive.

121 Mika Seppälä: Solved Problems on Optimization OPTIMIZATION Solution(contd) OPTIMIZATION

122 Mika Seppälä: Solved Problems on Optimization OPTIMIZATION Solution(contd) OPTIMIZATION We solve the above equation for x to find the critical points. By squaring both sides

123 Mika Seppälä: Solved Problems on Optimization OPTIMIZATION Solution(contd) OPTIMIZATION Then

124 Mika Seppälä: Solved Problems on Optimization OPTIMIZATION Solution(contd) OPTIMIZATION Using quadratic formula we find

125 Mika Seppälä: Solved Problems on Optimization OPTIMIZATION Solution(contd) OPTIMIZATION If we choose negative value for, then x would be negative which is not possible since it measures a distance. Therefore

126 Mika Seppälä: Solved Problems on Optimization OPTIMIZATION Solution(contd) OPTIMIZATION This value of x maximizes the distance since, In other words, the value of x that maximizes the distance is the solution of the equation

127 Mika Seppälä: Solved Problems on Optimization OPTIMIZATION Solution(contd) OPTIMIZATION By rewriting this equation, we obtain Since and, the equation above simplifies to.

128 Mika Seppälä: Solved Problems on Optimization OPTIMIZATION Solution(contd) OPTIMIZATION From the figure, we read that and. Hence, from which it follows that.


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