Linear programming: the best answer is at a corner

0:01A bakery makes x trays of muffins and y trays of cookies. Which mix earns the most profit?

0:09Linear programming finds the best value within limits, called constraints.

0:14Oven space: x plus y is at most 10.

0:19Prep hours: 2x plus y is at most 16.

0:24Trays cannot be negative, so x and y are at least 0.

0:29The points that meet every constraint form the feasible region.

0:34The profit, P equals 30x plus 20y dollars, is the objective function. We want it as large as possible.

0:43The corners, called vertices, are (0, 0), (8, 0), (6, 4) and (0, 10).

0:52(6, 4) is where the lines cross: 6 plus 4 is 10, and 2 times 6 plus 4 is 16.

1:01(0, 0) gives 0 dollars, and (8, 0) gives 30 times 8, or 240 dollars.

1:10At (6, 4), 30 times 6 is 180 and 20 times 4 is 80, so 260 dollars.

1:19(0, 10) gives 20 times 10, or 200 dollars.

1:25The largest is 260 dollars, at (6, 4).

1:31Why a corner? Every point on this line earns 120 dollars.

1:36Slide it outward and the profit grows. The last point it touches is (6, 4).

1:44A common mistake is checking only the corners on the axes. That gives 240 dollars and misses the crossing at (6, 4).

1:53So bake 6 trays of muffins and 4 trays of cookies, for 260 dollars.

2:00Recap: the best value is at a corner, so test every corner.