1. Linear Programming
A potter wants to make and sell serving bowls and plates. A bowl uses 5 pounds of clay. A plate uses 4 pounds of clay. The potter has 40 pounds of clay and wants to make at least 4 bowls. Let x be the number of bowls made and y be the number of plates made.

2. Linear Programming
x ≥ 4 Make at least 4 bowls y ≥ 0 Number of plates cannot be negative 5x + 4y ≤ 40 Can use up to 40 pounds Suppose the profit on a bowl is $35 and the profit on a plate is $30. The total profit P is given by: P = 35x + 30y.

3. Linear Programming How do we maximize the profits? Graph constraints
Evaluate at each vertex

4. Linear Programming Point 1 (4,0): P = 35(4) + 30(0) = 140 Min
Point 3 (4,5): P = 35(4) + 30(5) = 290 Max
$290 is the maximum profit by making 4 bowls and 5 plates.