- 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 60≤p≤150. It has found that the number of cars rented per day can be modeled by the linear function n(p)=750−5p. 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?