Section 4.4 Optimization and Modeling

Steps in Solving Optimization Problems Understand the problem: The first step is to read the problem carefully until it is clearly understood. Ask yourself: What is unknown? What are the given quantities? What are the given conditions?

Steps in Solving Optimization Problems Understand the problem Draw a diagram. In most problems it is useful to draw a diagram and identify the given and required quantities on the diagram. (It might also help to try special cases)

Steps in Solving Optimization Problems Understand the problem Draw a diagram. Introduce Notation. Assign a symbol to the quantity that is to be maximized or minimized (let’s call it Q for now). Also select symbols (a, b, c,…, x, y) for other unknown quantities and label the diagram with these symbols.

Steps in Solving Optimization Problems Understand the problem Draw a diagram. Introduce Notation. Write a formula. Express Q in terms of some of the other symbols.

Steps in Solving Optimization Problems Understand the problem Draw a diagram. Introduce Notation. Write a formula. Write Q as a function of one variable. Use the given information to find relationships (in the form of equations) among the variables. Use these equations to eliminate all but one of the variables.

Steps in Solving Optimization Problems Understand the problem Draw a diagram. Introduce Notation. Write a formula. Write Q as a function of one variable. Find the domain of this function

Steps in Solving Optimization Problems Understand the problem Draw a diagram. Introduce Notation. Write a formula. Write Q as a function of one variable. Find the domain of this function Find the global maximum or global minimum value of this function.

Example A resort wants to build a 8000 square foot rectangular pool. It is to have 40 feet of deck area along the short sides and 8 feet of deck area around the long sides. What size of plot will they need to satisfy these conditions without wasting space.

Example You have to enclose 3200 square feet of land (in a rectangle) using a fence on 3 sides that will cost $2 per foot, and a special wall on the fourth side that will cost $6 per foot. What are the dimensions of your enclosure that will minimize your cost?

Example A box with a square base and an open top needs to be constructed that will have a volume of 3200m3. What should be the dimensions of the box so that the surface area will be minimized? Hint: draw a picture of the situation

Example A rectangle is to be inscribed within a right triangle with a base of 3 and a height of 4. What is the largest rectangle that can be created?

Example A cylinder is to be created within a sphere of radius r. What should are the dimensions of the cylinder which maximize its volume?

Example A company that makes televisions has a revenue function, R(x) that gives the revenue in thousands of dollars for a given input x where x is measured in 100s of televisions. C(x) gives the cost associated in producing televisions in hundreds of dollars (x is still measured in units of 100 televisions) Interpret R’(50) = 2 and C’(50) = 10 Should they increase or decrease their production from x = 50?

Example Find the quantity q which maximizes profit if the total revenue, R(q), and total cost, C(q) are given in dollars by Does the value of q give you a local max or a local min? How can you tell?

Example A hotel if they charge $300 per night for a hotel room, they can rent out a total of 20 rooms. They find that for each $25 decrease in price, they can rent an additional room. How many rooms should they rent out to maximize their revenue? What is their maximum revenue?