AMTH122 Survey of Calculus with Applications

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AMTH122Survey of Calculus with Applications

• Optimization

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Homework Due Today

• Homework
• Topic 16 Exercise 1-12, 15,30,45,55

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Optimization Problem

• Best
• Smallest
• Most efficient
• etc
• Maximum or Minimum Questions
• Find critical points f(x) 0
• Check whether the function reach extreme value at
  those points

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Problem-Solving Strategy for Optimization Problems

• Identify the quantity to be maximized or
  minimized (the output) and the quantity or
  quantities on which the output quantity depends
  (the input).
• Sketch and label a picture if the problem is
  geometric in nature.
• Write the constraint equation
• Build a model for the quantity that is to be
  maximized or minimized.
• Rewrite the equation in Step 3 in terms of only
  one input variable.
• Identify the input interval.
• Apply optimization techniques to the final
  equation.
• Answer the question or questions posed in the
  problem.

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Simplest geometry model Minimum Perimeter for
Fixed Area

• The perimeter of a rectangle is 60 yards. What
  dimensions will result in maximum area?
• Step1 input variables
• w--- width
• l length
• output variables
• A --- area of rectangle
• Step2
• Step 3 2w2l 60 (yards)
• Step 4
• A l w l (30-l)
• 0ltl lt 30

• Step 5
• A(l) 30 2l 0
• l 15
• A(15) 125
• A(0) A(30) 0
• The area of rectangle is maximized if the
  dimensions are 15 yards by yards.

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The best dimension of a page

• A page is of 96 square inches. The top and bottom
  margins are 1 ½ inches, and the left and right
  margins are 1 inches. What dimensions of the page
  will result in maximum print area?
• Step1
• input variable length and width of the page
• Output variable to maximize print area
• Answer 12 inch X 8 inch

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Practice Best window shape

• A Norman window has the shape of a rectangle
  topped by a semicircle. If the area of the
  rectangular window is 50 square feet, what
  dimensions will minimize the perimeter of the
  window?

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Optimization in Business

• Revenue
• R revenue, p --- price, q quantity
• R p q
• Maximizing the revenue
• Determine the price to obtain the maximum revenue
• Maximizing profit
• Determine the price to obtain the maximum profit

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Maximum profit

• Lindsay owns a tattoo parlor. She charges p for
  a basic tattoo, the demand is given by
• D(p) -0.02 p2 0.03 p 1825
• per year. If it costs Lindsay 25 per tattoo,
  what p price generates the maximum profit?
• Variable Let p be the price, P be the profit
• Objective P(p) D(p)p 25D(p) -0.02 p3
  0.53 p2 1824.25 p 45625
• Constraint None
• Find the maximum
• P(p) -0.06 p2 1.06 p 1824.25 0
• p183.4248 or -165.7581
• P(183.4248) 183394
• Conclusion
• The maximum profit is around 183,394 when the
  price of tattoo is 183.4248

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Maximizing yield

• Marge is planning a casino bus trip. If 100
  people sign up, the cost is 300 per person. For
  each additional person above 100, the cost per
  person is reduced by 2 per person. To maximize
  Marges revenue, how many people should go on the
  trip? What is the cost per person.
• Variable let x be the number of the person, y be
  the cost per person
• Objective maximize the revenue R xy
• Constraint y 300 2(x-100)500-2x
• 100ltxlt250
• Find the maximum
• Conclusion To maximize Marges revenue 125
  people should go on the trip at a cost per person
  of 250 per person.

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Economic Model

• Jimmy takes tourists on swamp tours in southern
  Louisiana. A two-hour tour costs 25. The limit
  of the group size is 18. In order to promote the
  tours, he offers a discount to groups. Jimmy
  discounts the price per person by 1.25 for each
  person in the group over 10. For example, if a
  church group has 12 people that take the tour,
  the price per person would be 25 2(1.25)
  22.50.
• (a) What size group will produce the largest
  revenue for Jimmy, and what is the largest
  possible revenue for a tour?
• (b) If Jimmy limits the group size to 13 people,
  what is his maximum possible revenue for each
  group?
• (c) If you were Jimmy, would you set a limit on
  the number of people in each group?
• Variables
• Objectives Output quantity to be maximized
  Revenue for Jimmy Input quantity size of the
  group.

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Inventory Control

• Determining how many production runs should be
  done to minimize production costs
• n ---- number of production runs
• x ---- number of products in each run.
• Example A soda bottling company bottles 20000
  cases of lime sode each year. Each production
  run costs 1400, plus a storage cost of 18 per
  case. How many production runs should the company
  use to minimize inventory cost? How many cases
  are bottled in each production run?
• C(x) order cost storage cost
• 1400n x/218
• Ans x 1764 n11

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More Exercise

• Sammys Snowmobiles forecasts that they will sell
  50 snowmobiles each month. When Sammy orders the
  snowmobiles, he must pay 250 to place the order
  plus 7000 per snowmobile. It costs Sammy 20 a
  year to store a snowmobile.
• How many snowmobiles should Sammy order, and how
  often should he place orders, to minimize total
  annual costs?