Emily sells basic and deluxe prom dresses that she makes. She earns a profit of $20 per basic dress and $35 per deluxe dress. She must determine the number of each type of dress to sell to maximize her total profit given constraints on the fabric materials purchased. Selling 0 basic dresses and 60 deluxe dresses results in the highest profit of $2100.

1. (1, 2) (1, 4) (5, 8) (5, 2) Maximum value = 11 at (5,2) Minimum value = -5 at (1, 4) x y 3x – 2y 1 2 3(1)-2(2) = -1 5 2 3(5)-2(2) = 11 5 8 3(5)-2(8) = -1 1 4 3(1)-2(4) = -5

2. What do you need to make a paper airplane? Paper Money (to buy the paper) Time Resources

3. Emily sells prom dresses that she makes herself. For each basic dress she sells, she earns a profit of $20, and each deluxe dress she sells earns a profit of $35. To make each basic dress, she must use 2 yards of cotton fabric and 2 yards of silk; each deluxe dress requires 3 yards of cotton and 1 yard of silk. She has purchased a total of 180 yards of cotton fabric and 100 yards of silk. Determine the number of basic dresses and the number of deluxe dresses she must sell to maximize her profit. Resources: Cotton Fabric Silk Fabric

4. Emily sells prom dresses that she makes herself. For each basic dress she sells, she earns a profit of $20, and each deluxe dress she sells earns a profit of $35. To make each basic dress, she must use 2 yards of cotton fabric and 2 yards of silk; each deluxe dress requires 3 yards of cotton and 1 yard of silk. She has purchased a total of 180 yards of cotton fabric and 100 yards of silk. Determine the number of basic dresses and the number of deluxe dresses she must sell to maximize her profit. Variables: number of Basic dresses number of Deluxe dresses B D

5. Emily sells prom dresses that she makes herself. For each basic dress she sells, she earns a profit of $20, and each deluxe dress she sells earns a profit of $35. To make each basic dress, she must use 2 yards of cotton fabric and 2 yards of silk; each deluxe dress requires 3 yards of cotton and 1 yard of silk. She has purchased a total of 180 yards of cotton fabric and 100 yards of silk. Determine the number of basic dresses and the number of deluxe dresses she must sell to maximize her profit. Objective function: 20B + 35D

6. To make each basic dress, she must use 2 yards of cotton fabric and 2 yards of silk; each deluxe dress requires 3 yards of cotton and 1 yard of silk. She has purchased a total of 180 yards of cotton fabric and 100 yards of silk. Constraints Variables Basic Dresses Deluxe Dresses Total Resources Cotton Silk

7. To make each basic dress, she must use 2 yards of cotton fabric and 2 yards of silk; each deluxe dress requires 3 yards of cotton and 1 yard of silk. She has purchased a total of 180 yards of cotton fabric and 100 yards of silk. Constraints 180 100 2 2 3 1 B B D D 2B + 3D  180 2B + D  100 B ≥ 0 D ≥ 0 Variables Basic Dresses Deluxe Dresses Total Resources Cotton Silk

8. 2B + 3D  180 2B + D  100 B ≥ 0 D ≥ 0 20 40 60 80 100 Basic dresses Deluxe dresses 100 80 60 40 20 (0,100) (90, 0) (50, 0) (0, 60) Vertices: (0, 60) (50, 0) (0, 0) (30, 40)

9. Values of vertices in objective function (0, 60) (50, 0) (0, 0) (30, 40) 20B + 35D (B, D) 20(0) + 35(60) = 2100 20(50) + 35(0) = 1000 20(0) + 35(0) = 0 20(30) + 35(40) = 2000 Determine the number of basic dresses and the number of deluxe dresses she must sell to maximize her profit. Sell 0 Basic and 60 Deluxe Dresses for a maximum profit of $2100

10. (-5,-2) (3, 2) (1,-2) Maximum value = 10 at (3,2) Minimum value = -18 at (-5, -2) x y 4x – y 3 2 4(3)-(2) = 10 1 -2 4(1)-(-2) = 6 -5 -2 4(-5)-(-2) = -18