- 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 [latex]f'(x) = 0[/latex] or [latex]f'(x)[/latex] is undefined
- Key to finding potential extrema
- Constraint Equations:
- Used to express the problem in terms of a single variable
- Often in the form [latex]g(x,y) = k[/latex], where [latex]k[/latex] is a constant
- Domain Analysis:
- Crucial for determining where to search for extrema
- Consider endpoints for closed intervals
- Analyze behavior as [latex]x \to \infty[/latex] or [latex]x \to -\infty[/latex] 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 [latex]24[/latex] in. by [latex]36[/latex] 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 [latex]x[/latex] be the side length of each square and write the volume of the open-top box as a function of [latex]x[/latex]. Determine the domain of consideration for [latex]x[/latex].
Suppose the island is [latex]1[/latex] mi from shore, and the distance from the cabin to the point on the shore closest to the island is [latex]15[/latex] mi. Suppose a visitor swims at the rate of [latex]2.5[/latex] mph and runs at a rate of [latex]6[/latex] mph. Let [latex]x[/latex] 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 [latex]p[/latex] dollars per day, where [latex]60\le p\le 150[/latex]. It has found that the number of cars rented per day can be modeled by the linear function [latex]n(p)=750-5p[/latex]. How much should the company charge each customer to maximize revenue?
Modify the area function [latex]A[/latex] if the rectangle is to be inscribed in the unit circle [latex]x^2+y^2=1[/latex]. What is the domain of consideration?