Linear Programming

Graph the following system of inequalities

Graph the system

Another example

Graphing Inequalities

More graphing...

Linear Programming A method used to find optimal solutions such as maximum or minimum profits Steps: 1.Assign variables 2.Determine the constraints (inequalities) 3.Find the feasible region (area of solution) 4.Determine the vertices of feasible region 5.Plug those values into the profit equation(also called objective function)

Finding Max or Min

Example Mr. Farmer wants to plant some corn and wheat and he gets the following statistics from the US Bureau of Census CropYield per acreAvg Price Corn113.5 bu$3.15/bu Soybeans34.9 bu$6.80 bu Wheat35.8 bu$4.45 bu Cotton540 lb$0.759/lb Rice, rough5621 lb$0.0865/lb

Example continued Mr. Farmer can have no more 120 acres of corn and wheat At least 20 and no more than 80 acres of corn At least 30 acres of wheat How many acres of each crop should Mr. Farmer plant to maximize the revenue from his harvest?

Working through the problem.. Assign Variables X=acres of corn and y=acres of wheat List the constraints

Graph it

Example continued List the vertices Determine Profit equation Which would yield the most?

Another Ex. A snack bar cooks and sells hamburgers and hot dogs during football games. To stay in business, it must sell at least 10 hamburgers but cannot cook more than 40. It must also sell at least 30 hot dogs but cannot cook more than 70. The snack bar cannot cook more than 90 items total. The profit on a hamburger is $0.33 and on a hot dog it is $0.21. How many of each item should it sell to make the maximum profit? Profit Equation: __________________________ Constraints: Answer: _________________________

Another Example As a receptionist for a veterinarian, Sue scheduled appointments. She allots 20 minutes for a routine office visit and 40 minutes for surgery. The vet can not do more than 6 surgeries per day. The office has 7 hours available for appointments. If an office visits costs $55 and most surgeries costs $125, find a combination of office visits and surgeries that will maximize the income the veterinarian practice receives per day.

What do you know… Assign variables x=number of office visits y=number of surgeries Constraints: 7 hours needs to be in terms of minutes

Continued… Graph and determine coordinates of the vertices You should get (0,0) (0,6)(9,6)(21,0)

Continued…. Determine the profit equation $55v + $125s = P Test the points Highest profit would be when there are 6 surgeries and 9 visits