Optimization Problems November 29th

Pre-Lesson Activity Write equations for the following statements: A machine holds, at most 240 cans of drinks Sales from the vending machine show that at least 2 cans of juice are sold for each can of pop.   No more than 10 friends at a party Chocolate bars are sold for $2.50 each and M&M’s are sold at $1.00 per pack. You have 10 dollars in your pocket. Melanie is studying for two exams, Biology (x) and Math (y). She needs to spend at least twice as much time studying for biology than for math. She wants to spend no more than 18 hours studying for either.

From 6.1 to 6.3 We learned to graph linear inequalities like this: 6 > 2x + 3y x ∈ W y ∈ W

From 6.1 to 6.3 We learned to graph linear inequalities like this: 6 > 2x + 3y x ∈ W y ∈ W #1 Graph the linear equation #2 Is the line dotted or solid? #3 Where is the solutions region? #4 What are the restrictions on the solutions region?

From 6.4 to 6.6 We will learn to apply these skills to optimization problems

Optimization Problems Optimizations Problems occur when we are trying to optimize our product (e.g. revenues, savings etc.) for a certain scenario. In other words, when we are trying to find the best solution out of many different scenarios.

Example 1 You are trying to create a fruits basket. Each basket contains at least 5 apples and at least 6 oranges. Apples cost $0.20 and oranges cost $0.35. The budget allows no more than 7 dollars, for all fruits in the basket.

Objective Function In an optimization problem, the equation that represents the relationship between the two variables in the system of linear inequalities and the quantity to be optimized.

Objective Function What am I trying to optimize in this problem? How do my two variables (number of oranges and apples) relate to this equation?

Example 2 A fast-food concession stand sells hotdogs and hamburgers. Daily sales can be as high as 300 hamburgers and hot dogs combined. The stand has room to stock no more than 200 hot dogs and no more than 150 hamburgers. Hot dogs are sold for $3.25, and hamburgers are sold for $4.75.

Example 2 Creating a model means: Writing the system of inequalities, with restrictions Graphing the system of inequalities. Writing an objective function

Homework Page 330 #3-7