Do Now The cost of renting a pool at an aquatic center is either $30 an hr. or $20 an hr. with a $40 non refundable deposit. Use algebra to find for how many hours the cost of renting the pool is the same for both plans. Let x = # of hrs 30x = 20x + 40 10x = 40 X = 4 LT: I can use constraints to find possible locations on a graph to maximize profit and minimize cost.

Entry Task Graph the inequality 3x + 4y < 12 3(0) + 4y < 12 y < 3 so (0,3) And… 3x + 4(0) < 12 3x < 12 X < 4 so (4,0) LT: I can use constraints to find possible locations on a graph to maximize profit and minimize cost.

3-4: Linear Programming LT: I can use constraints to find possible locations on a graph to maximize profit and minimize cost. LT: I can use constraints to find possible locations on a graph to maximize profit and minimize cost.

Linear Programming Businesses use linear programming to find out how to maximize profit or minimize costs. Most have constraints on what they can use or buy. LT: I can use constraints to find possible locations on a graph to maximize profit and minimize cost.

Find the minimum and maximum value of the function f(x, y) = 3x - 2y. We are given the constraints: y ≥ 2 1 ≤ x x ≤ 5 y ≤ x + 3 LT: I can solve problems using linear programming

Given the constraints (equations) y ≥ 2 1 ≤ x x ≤ 5 y ≤ x + 3 Graph these lines on your graph paper. What do you notice? Turn to your elbow partner #1 first, then #2 What does what you are looking at represesnt? LT: I can use constraints to find possible locations on a graph to maximize profit and minimize cost.

What if….. What if I said that one of those points in the shaded area represents the maximum profit and minimum costs for the profit equation f(x,y) = 3x – 2y. Then I said to find them. Where would you start? Why? On your own…. Elbow partner, #2 first then #1, Share out LT: I can use constraints to find possible locations on a graph to maximize profit and minimize cost.

Corner Point Principle Which point is optimal? Any point in feasible region will satisfy constraint equation, but which will maximize profit equation? Corner Point Principle In LP problem, the maximal value for profit always corresponds to a corner point on feasible region ( 0, 30 ) ( 12, 0 ) ( 0, 0 ) LT: I can use constraints to find possible locations on a graph to maximize profit and minimize cost.

1 ≤ x x ≤ 5 y ≥ 2 y ≤ x + 3 So how do we find those 4 points? 8 7 6 5 LT: I can use constraints to find possible locations on a graph to maximize profit and minimize cost.

The vertices of the quadrilateral formed are: Linear Programming The vertices of the quadrilateral formed are: (1, 2) (1, 4) (5, 2) (5, 8) Plug these points into the function f(x, y) = 3x - 2y LT: I can use constraints to find possible locations on a graph to maximize profit and minimize cost.

Calculating Maximum Profit f(x, y) = 3x - 2y f(1, 2) = 3(1) - 2(2) = 3 - 4 = -1 f(1, 4) = 3(1) - 2(4) = 3 - 8 = -5 f(5, 2) = 3(5) - 2(2) = 15 - 4 = 11 f(5, 8) = 3(5) - 2(8) = 15 - 16 = -1 LT: I can use constraints to find possible locations on a graph to maximize profit and minimize cost.

Linear Programming f(1, 4) = -5 minimum f(5, 2) = 11 maximum LT: I can use constraints to find possible locations on a graph to maximize profit and minimize cost.

Example: Linear Programming problem Example: Sarah makes bracelets and necklaces to sell at a craft store. Each bracelet makes a profit of $7, takes 1 hour to assemble, and costs $2 for materials. Each necklace makes a profit of $12, takes 2 hour to assemble, and costs $3 for materials. Sarah has 48 hours available to assemble bracelets and necklaces. If she has $78 available to pay for materials, how many bracelets and necklaces should she make to maximize her profit? To formulate this as a linear programming problem: 1. Identify the variables. 2. Write the objective function. 3. Write the constraints. Example continued Example: Linear Programming problem

1. Let x = the number of bracelets Sarah makes Example continued: 1. Let x = the number of bracelets Sarah makes Let y = the number of necklaces Sarah makes 2. Express the profit as a function of x and y. p = 7x + 12y Function to be maximized 3. Express the constraints as inequalities. Cost of materials: 2x + 3y  78. Time limitation: x + 2y  48. Since Sarah cannot make a negative number of bracelets or necklaces, x  0 and y  0 must also hold. Example continued Example continued

Maximize p = 7x + 12y subject to the constraints Example continued: Maximize p = 7x + 12y subject to the constraints 2x + 3y  78, x + 2y  48, x  0, and y  0. y x (0, 24) (12, 18) z = 300 z = 288 (0, 0) (39, 0) z = 0 z = 273 Sarah should make 12 bracelets and 18 necklaces for a maximum profit of $300. Example continued

Summary Find the minimum and maximum values by graphing the inequalities and finding the vertices of the polygon formed. Substitute the vertices into the function and find the largest and smallest values. LT: I can use constraints to find possible locations on a graph to maximize profit and minimize cost.

Homework Homework P. 160– #10, 13, 14, 16 LT: I can use constraints to find possible locations on a graph to maximize profit and minimize cost.

Setting Up Mixture Problems Example 1: Toy manufacturer can produce skateboards and dolls. Both require the precious resource of plastic, of which there are 60 units available. Skateboards take five units of plastic and make $1 profit. Dolls take two units of plastic and make $0.55 profit. What is the number of dolls and skateboards the company can produce to maximize profit? LT: I can use constraints to find possible locations on a graph to maximize profit and minimize cost.

Setting Up Mixture Problems First identify components of the problem: 1. Resources – Plastic (60 available) 2. Products – Skateboards & Dolls 3. Recipes – Skateboards (5 units), Dolls (2 units) 4. Profits – Skateboards ($1.00), Dolls $0.55) 5. Objective – Maximize profit LT: I can use constraints to find possible locations on a graph to maximize profit and minimize cost.

Writing Equations 2 Groups of Equations: - Profit Equation Resources Plastic (60) Profit Products Skateboards (x units) 5 $1.00 Dolls (y units) 2 $0.55 2 Groups of Equations: - Profit Equation - Constraints equations Profit Equation - total profit given number of units produced P = 1x + 0.55y Constraints – usually inequalities 5x + 2y ≤ 60 x > 0 y > 0 With these, create Feasible Region LT: I can solve problems using linear programming

Mixture Problems Feasible Region – region which consists of all possible solution choices for a particular problem Using the constraint equations we get the following graph: Constraints: 5x + 2y ≤ 60 LT: I can use constraints to find possible locations on a graph to maximize profit and minimize cost.

Corner Point Principle Plug in corner points to profit formula: Corner point (0,30) is the optimal point Therefore the optimal solution would be to produce 0 skateboards and 30 dolls ( 0, 30 ) Point Calculation of Profit Formula $1.00x + $0.55y = P (0, 0) $1.00 (0) + $0.55 (0) = $0.00 (0, 30) $1.00 (0) + $0.55 (30) = $16.50 (12, 0) $1.00 (12) + $0.55 (0) = $12.00 ( 12, 0 ) ( 0, 0 ) LT: I can use constraints to find possible locations on a graph to maximize profit and minimize cost.

LT: I can use constraints to find possible locations on a graph to maximize profit and minimize cost.

We are given the constraints: y ≥ -x + 2 y ≤ x + 2 y ≥ 2x -5 Example 2 Find the minimum and maximum value of the function f(x, y) = 4x + 3y We are given the constraints: y ≥ -x + 2 y ≤ x + 2 y ≥ 2x -5 LT: I can use constraints to find possible locations on a graph to maximize profit and minimize cost.

y ≥ 2x -5 6 5 4 y ≥ -x + 2 3 2 1 1 2 3 4 5 LT: I can use constraints to find possible locations on a graph to maximize profit and minimize cost.

Vertices f(x, y) = 4x + 3y f(0, 2) = 4(0) + 3(2) = 6 LT: I can use constraints to find possible locations on a graph to maximize profit and minimize cost.

Linear Programming f(0, 2) = 6 minimum f(4, 3) = 25 maximum LT: I can use constraints to find possible locations on a graph to maximize profit and minimize cost.

LT: I can use constraints to find possible locations on a graph to maximize profit and minimize cost.