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Applied Optimization Problems: Fresh Take

  • Tackle business problems to find the best ways to increase profits, minimize costs, or maximize revenue
  • Use optimization methods to solve problems involving geometry

Solving Optimization Problems

The Main Idea 

  • Optimization Goal:
    • Find maximum or minimum values of functions under certain constraints
  • Types of Intervals:
    • Closed and bounded: Guaranteed to have absolute extrema
    • Open or unbounded: May or may not have absolute extrema
  • Critical Points:
    • Points where f(x)=0 or f(x) is undefined
    • Key to finding potential extrema
  • Constraint Equations:
    • Used to express the problem in terms of a single variable
    • Often in the form g(x,y)=k, where k is a constant
  • Domain Analysis:
    • Crucial for determining where to search for extrema
    • Consider endpoints for closed intervals
    • Analyze behavior as x or x for unbounded intervals
  • Applications:
    • Geometry: Maximizing areas or volumes, minimizing surface areas
    • Business: Maximizing revenue or profit, minimizing costs
    • Physics: Minimizing energy, maximizing distances

In the Learn It pages, we gave the example: “An open-top box is to be made from a 24 in. by 36 in. piece of cardboard by removing a square from each corner of the box and folding up the flaps on each side. What size square should be cut out of each corner to get a box with the maximum volume?”

Suppose the dimensions of the cardboard are now 20 in. by 30 in. Let x be the side length of each square and write the volume of the open-top box as a function of x. Determine the domain of consideration for x.

Suppose the island is 1 mi from shore, and the distance from the cabin to the point on the shore closest to the island is 15 mi. Suppose a visitor swims at the rate of 2.5 mph and runs at a rate of 6 mph. Let x denote the distance the visitor will run before swimming, and find a function for the time it takes the visitor to get from the cabin to the island.

A car rental company charges its customers p dollars per day, where 60p150. It has found that the number of cars rented per day can be modeled by the linear function n(p)=7505p. How much should the company charge each customer to maximize revenue?

Modify the area function A if the rectangle is to be inscribed in the unit circle x2+y2=1. What is the domain of consideration?