Applied Optimization Problems: Fresh Take

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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 $x \to \infty$ or $x \to -\infty$ 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 $x$ .

The volume of the box is $L \cdot W \cdot H$ .

$V(x)=x(20-2x)(30-2x)$ . The domain is $[0,10]$ .

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.

The time $T=T_{\text{running}}+T_{\text{swimming}}$ .

$T(x)=\frac{x}{6}+\frac{\sqrt{(15-x)^2+1}}{2.5}$

A car rental company charges its customers $p$ dollars per day, where $60\le p\le 150$ . 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?

$R(p)=n \times p$ , where $n$ is the number of cars rented and $p$ is the price charged per car.

The company should charge $$75$ per car per day.

Modify the area function $A$ if the rectangle is to be inscribed in the unit circle $x^2+y^2=1$ . What is the domain of consideration?

If $(x,y)$ is the vertex of the square that lies in the first quadrant, then the area of the square is $A=(2x)(2y)=4xy$ .

$A(x)=4x\sqrt{1-x^2}$ . The domain of consideration is $[0,1]$ .