Warm Up 2/19

1. Warm Up 2/19 1) Solve by elimination. 8x + 14y = 4 -6x – 7y = -10 2) Solve by elimination 3x – 2y = 2 5x – 5y = 10 3) Solve by substitution. x = y + 11 2x + y = 9 4) Determine whether the function is linear, quadratic, or exponential.

2. EQ: Can the solutions of a system of linear inequalities be listed explicitly? Lesson 3: Linear Programming Baking a tray of corn muffins takes 4 cups of milk and 3 cups of flour. A tray of bran muffins takes 2 cups of milk and 3 cups of four. A baker has 16 cups of milk and 15 cups of flour. He makes $3 profit per tray of corn muffins and $2 profit per tray of bran muffins. How many trays of each type of muffin should the baker make to maximize his profit? Objective: X = Y = Constraints:

3. Objective: X = Y = Constraints:

4. List all the vertices (corner points) of the feasible region: These points represent the largest values of each combination of muffins that can be made within the constraints. Remember our objective function is to maximize profit. Where did we represent this? Which one of the vertices gives the baker its maximum profit?

5. List all the vertices (corner points) of the feasible region: These points represent the largest values of each combination of muffins that can be made within the constraints. Remember our objective function is to maximize profit. Where did we represent this? Which one of the vertices gives the baker its maximum profit?

6. Linear Programming is used every day to maximize their profits and minimize their costs. Steps for Linear Programming: • Define the unknowns (variables) • Write an objective function based on profits or cost • Write constraints (inequalities) based on limitations • Graph the constraints to find feasible vertices • Use the vertices and the objective function to either maximize the profit or minimize the cost

7. Use the vertices to find the maximum and minimum values Examples: • Given the following constraints, maximize and minimize the value of z = -0.4x + 3.2y

8. Examples: 2. Find the maximal and minimal value of z = 3x + 4y subject to the following constraints:

9. Let’s practice setting some up on our own. For each problem below, write an objective function and inequalities to represent the constraints. • Max is stenciling wooden boxes to sell at a fair. Small boxes take 2 hours to stencil and large boxes take 3 hours. Profit for a small box is $20 and for a large is $30. Max has no more than 30 hours of stenciling time but wants to make at least 12 boxes to sell. How many of each size box should he try to sell to maximize his profit? Objective: X = Y = Constraints:

10. Let’s practice setting some up on our own. For each problem below, write an objective function and inequalities to represent the constraints. 2. Bob builds tool sheds. He uses 10 sheets of dry wall and 15 studs for a small shed and 15 sheets of dry wall and 45 studs for a large shed. He has available 60 sheets of dry wall and 135 studs. If Bob makes $390 profit on a small shed and $520 on a large shed, how many of each type of building should Bob build to maximize his profit? Objective: X = Y = Constraints:

11. Let’s practice setting some up on our own. For each problem below, write an objective function and inequalities to represent the constraints. 3. The Cruiser Bicycle Company makes two styles of bicycles: the Traveler, which sells for $300 and the Tourister, which sells for $600. Each bicycle has the same frame and tires, but the assembly and painting time required for the Traveler is only 1 hour, while it is 3 hours for the Tourister. There are 30 frames and 360 hours of labor available for production. Lastly, to make the company quota for the month, they need to sell at least 25 of each bike. How many bicycles of each model should be produced to maximize revenue? Objective: X = Y = Constraints:

12. Let’s graph the bike problem: Objective: Constraints:

13. Ali formulated the following problem: • An electronics company is manufacturing e-book readers. A basic model takes 4 hours and $40 to make. A touch screen model takes 6 hours and $120 to make. The company has 10 employees working 12-hour days. The daily operating budget is $1920 per day for materials. The company would like at least 3 basic models and at least 8 touch screen models produced each day. How many of each model should the company produce to minimize their daily cost? x = # basic models Objective: minimize cost • y = # touch screen models C(x) = 120x + 120y • Constraints: • Hours: 4x + 6y < 120 Money: 40x + 120y < 1920 #basics: x > 8 #touch screens: y > 3

14. x = # basic models Objective: minimize cost • y = # touch screen models C(x) = 120x + 120y • Constraints: • Hours: 4x + 6y < 120 Money: 40x + 120y < 1920 #basics: x > 8 #touch screens: y > 3 Graph the constraints:

15. How many of each model should the company produce to minimize their daily cost? What will that minimum cost be?

16. Damon began solving the following problem: • The B&W Leather Company want to add handmade belts and wallets to its product line. Each belt nets the company $18 in profit, and each wallets nets $12. Both belts and wallets require cutting and sewing. Belts require 2 hours of cutting time and 6 hours of sewing time. Wallets require 3 hours of cutting time and 3 hours of sewing time. If the cutting machine is available 12 hours per week and the sewing machine is available 18 hours per week, how many belts and wallets will produce the most profit within the constraints? Here is how he formulated the problem: x = # belts Objective Function: maximize profit y = # wallets P(x) = 18x + 12y Constraints: Cutting: 2x + 3y ≤ 12 Sewing: 6x + 3y ≤ 18 x ≥0, y ≥ 0

17. Here is how he formulated the problem: x = # belts Objective Function: maximize profit y = # wallets P(x) = 18x + 12y Constraints: Cutting: 2x + 3y ≤ 12 Sewing: 6x + 3y ≤ 18 x ≥0, y ≥ 0 And his graph looks like this: According to this information, how many belts and wallets should the company make to maximize their profits? What will their profit be?