Sec 4 5 Optimization Applied MaxMin Problems 1

Sec 4. 5 Optimization (Applied Max/Min Problems) 1

Optimization Problems Given: A task. Objective: Find the “best” way to complete the task. Depends on our concern. Specific Examples: (1) Cheapest way to do a certain thing; (2) Fastest way to do a certain thing; (3) To obtain the largest among a group of things… 2

Example (1) Manufacturing a can Task: To manufacture a (cylindrical) can with a given capacity, say 500 m. L. “Best”: We want to use the least amount of material. 3

Example (2) Making a window Task: To manufacture a window with a given shape and perimeter. “Best”: We want the window to admit the greatest possible amount of light. Note: Extra (specific) information will be provided … 4

Example (3) Fitting a rectangle inside a region Task: To place a rectangle inside a given region. “Best”: We want the rectangle to be as large as possible (in terms of area). 5

Example (4) Finding a ladder Task: To find a ladder to reach over a fence to a building. “Best”: We want the shortest ladder that can do the job. 6

Example (5) Offshore oil-well problem Task: To build pipes for oil transportation. “Best”: We want the cost to be as low as possible. 7

Example (6) Rescue Operation Task: To get to a plane crash site. “Best”: We want to get there as quickly as possible. 8

Example (1) Manufacturing a can Task: To manufacture a (cylindrical) can with a given capacity, say 500 m. L. Height h Such cans come in all shapes and sizes. They differ from one another in dimensions. Base radius r “Best”: We want to use the least amount of To find the absolute minimum of material. Amount of material, A. 9

Example (2) Making a window Task: To manufacture a window with a given shape and perimeter. Extra information: The perimeter, say 4 m. Such windows come in all sizes. They differ from one another in dimensions. “Best”: We want the window to admit the greatest possible amount of light. To find the absolute maximum of Area of the window, A. r h 10

Method of Solution (1) Read the question carefully, understand the task involved, and the many possible ways of completing the task. (2) Identify the quantity that you want to maximize (or minimize) while completing the task. (3) Express this quantity as a function of some suitably chosen parameter that represents the possible ways of completing the task. (4) Find the absolute maximum (or minimum) of your quantity. 11

Example (3) Fitting a rectangle inside a region Task: To place a rectangle inside a given region. Extra information: The specifics about the region, say … y = 8 – x 2 Such rectangles come in all shapes and sizes. They differ from one another in dimensions … … or, the “Best”: We want the location of their rectangle to be as corners. large as possible (in terms of area). (x, y) y=0 To find the absolute maximum of Area of the rectangle, A. 12

Example (4) Finding a ladder Task: To find a ladder to reach over a fence to a building. Extra information: The specifics about the fence, say … Such ladders come in all sizes. They differ from one another in inclination … … or, the “Best”: We want the angles they shortest ladder that make with the ground. L 2 m can do the job. 2 m To find the absolute minimum of Length of the ladder, L. 13

Example (5) Offshore oil-well problem Task: To build pipes for oil transportation. “Best”: We want the cost to be as low as possible. Such pipes come in many forms. They differ from one another in where the junction is located … To find the absolute minimum of cost, C. 3 km Extra information: The specifics about various costs, say … Underwater pipe costs $ 400, 000 per km Pipe on land costs $ 300, 000 per km Location of oil well: 3 km off shore Location of factory: 6 km down x km 6 km 14

Example (6) Rescue Operation Task: To get to a plane crash site. “Best”: We want to get there as quickly as possible. Many paths lead to the plane. They differ from one another in where the “take-off point” is located … To find the absolute minimum of time, T. Extra information: The specifics about speeds, say … Speed on sand 50 km/h 5 km Speed on land 130 km/h Location of plane: 5 km off road “Take-off Location of hospital: 8 km down point” x km 8 km 15