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Optimization: Applied max/min problems

Optimization: Applied max/min problems. Analyze using earlier techniques Two main cases: Problems reducing to max/min of functions on closed bounded intervals Problems reducing to max/min of functions on other types of intervals. Squares vs. rectangles.

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Optimization: Applied max/min problems

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  1. Optimization: Applied max/min problems • Analyze using earlier techniques • Two main cases: • Problems reducing to max/min of functions on closed bounded intervals • Problems reducing to max/min of functions on other types of intervals

  2. Squares vs. rectangles • Find the dimensions of a rectangle with perimeter 32 m whose area is as large as possible.

  3. Recipe for applied max/min problems • Draw picture and label relevant quantities • Find a formula for quantity to be optimized • Eliminate variables to get function of one variable • Find functions max/min using earlier techniques

  4. Boxers • A box which is open on top is made from a 16 inch by 30 inch cardboard by cutting out squares of equal size from the four corners and bending up the sides. What size should the squares be to obtain a box with largest possible volume?

  5. Campbell’s Soup • Find the dimensions of acan with the smallest possible surface area that encloses a volume of 16 cm3 of chicken noodle soup.

  6. Profit = Revenue - Cost • Microsoft sells PowerPoint at a price of $100 per program. If the daily production cost in dollars for x copies of the program is C(x) = 100,000 + 50x + 0.0025x2and if the daily production capacity is at most 7000 copies, how many copies must be manufactured and sold to maximize the profit? • Should Microsoft expand the daily production capacity?

  7. Bay Watch • A pretty girl is drowning. She is 100 m down-shore from a lifeguard and she is 9 m offshore. The lifeguard can swim 1 m/sec and he can run 2 m/sec. What path should he take to get to the girl as quickly as possible?

  8. Different choices for optimization function • Find a point on the curve y = x2 that is closest to the point (18,0).

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