3.7 Optimization Problems

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3.7 Optimization Problems Applied Minimum and Maximum Problems Problem-Solving Strategy is written on Page. 206. Assign Symbols (choose variables) Primary Equations (objective function) Secondary Equations Domain (interval) Solve using calculus (optimize the function

Example 1: A piece of wire of length L is bend into the shape of a rectangle. Which dimensions produce the rectangle of maximum area?

Example 2: Your task is to build a road joining a ranch to a highway that enables drivers to reach the city in the shortest time. How should this be done if the speed limit is 60km/h on the road and 110 km/h on the highway? The perpendicular distance from the ranch to the highway is 30 km, and the city is 50km down the highway.

Example 3: All units in a 30-unit apartment building are rented out when the monthly rent is set at r = $1000/month. A survey reveals that one unit becomes vacant with each $40 increase in rent. Suppose that each occupied unit costs $120/month in maintenance. Which rent r maximizes monthly profit?

Example 4: Design a cylindrical can of volume 900 cm3 so that it uses the least amount of metal. In other words, minimize the surface area of the can (including its top and bottom).

Example 5: Is it possible to design a cylinder of volume 900 cm3 with the largest possible surface area?