Linear Programming

1. Linear Programming • set up and solve linear programming problems to find real life minimums and maximums.

2. Linear Programming Hints • To write the constraints and objective function: • Identify the variables. • Write the objective function • Identify each constraint. Then write an inequality for each. • Start with x> 0 and y> 0 (usually).

3. Example 1 • The West Hartford Senior Center is trying to establish a transportation system of small and large vans. It can spend no more than $100,000 for both sizes of vehicles and no more than $500 per month for maintenance. The WHSC can purchase a small van, which carries up to 7 passengers, for $10,000 and maintain it for $100 per month. The large vans, which carry up to 15 passengers, cost $20,000 each and can be maintained for $75 per month. How many of each type of van should they purchase if they want to maximize the number of passengers? Initial purchase cost # of passengers Monthly maintenance P = 7s + 15l

4. 10 8 6 4 2 0 Example 1 The vertices that should be tested are (0,5), (2,4), and (5,0). # of large vans, l The WHSC should buy 5 large vans which can hold 75 passengers. 0 2 4 6 8 10 # of small vans, s

5. Example 2 • Pancakes Waffles 3 cups Bisquick 2 cups Bisquick 1 cup Milk 2 cups Milk 2 Eggs 2 Eggs Serves 6 Serves 5 • You have 24 cups of Bisquick, 18 cups of milk, and 20 eggs. If you want to feed as many people as possible, how many batches of each should you make? Servings Bisquick Milk Eggs

6. 10 8 6 4 2 0 Example 2 The vertices that should be tested are (0,9), (2,8), (4,6), and (8,0). # of batches of waffles, w Make 4 batches of pancakes and 6 batches of waffles to feed 54 people. 0 2 4 6 8 10 # of batches of pancakes, p

7. Example 3 • Kayla works no more than 20 hours per week during the school year. She is paid $10 an hour for tutoring Geometry students and $7 an hour for babysitting. She wants to spend at least 3 hours but no more than 8 hours a week tutoring. Find Kayla’s maximum weekly earnings. Earnings Total hours worked

8. 20 16 12 8 4 0 Example 3 The vertices that should be tested are (3,17) and (8,12). # of hrs spent babysitting, b Kayla should tutor for 8 hours and babysit for 12 hours. 0 2 4 6 8 10 # of hrs spent tutoring, t

9. Example 4 • As part of your weight training regimen, you want to consume lean sources of protein. You want to consume at least 300 Calories a day from at least 48 grams of protein. One ounce of chicken provides 35 Calories and 8.5 g of protein. One ounce of tofu provides 20 Calories and 2.5 g of protein. Your local supermarket charges $5 a pound for chicken and $2.50 a pound for tofu. How much of each food should you eat each day if you want to meet your requirements with the lowest cost? What is this daily cost? Calories Price Paid Protein

10. 1.0 0.8 0.6 0.4 0.2 0 Example 4 The vertices that should be tested are approx. (0, 1.2), (0.16, 0.65), and (.536,0). # of lbs of tofu, t You should eat about 0.16 pounds of chicken and 0.65 pounds of tofu, which will cost about $2.43 per day. 0 0.2 0.4 0.6 0.8 1.0 # of lbs of chicken, c