Solve: "Baking a tray of corn muffins takes 4 cups of milk and 3 cups of wheat flower.

Baking a tray of bran muffins takes 2 cups of milk and 3 cups of wheat flour. A baker has 16 cups of milk and 15 cups of wheat 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 maxamize his profit?"

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We know from enunciation that baking a tray of corn muffins takes 4 cups of milk and 3 cups of wheat flour and the resulted profit is $3. And baking a tray of bran muffins takes 2 cups of milk and 3 cups of wheat flour and the profit is $2.

From the amount of 16 cups of milk the baker can make 4 trays of corn muffins or 6 trays of bran muffins.

From the amount of 15 cups of wheat flour the baker can bake the same number of trays: 5 trays of corn muffins or 5 trays of bran muffins.

We'll impose the following constraints:

- for baking x trays of corn muffin, the profi made is: $3x

- for baking y trays of bran muffin, the profit made is $2y

- 3x + 2y < 15 (the smaller amount of wheat flour)

We'll express y with respect to x. For this reason, we'll subtract 3x both sides:

2y < 15 - 3x

We'll divide by 2:

y < (15 - 3x)/2

For x = 1

y < (15 - 3)/2

y< 6

The profit made is:

3x + 2y = 3*1 + 2*6 = 3 + 12 = $15

For x = 2

y < (15 - 6)/2

y < 9/2

y < 4.5

Since y is integer, then y = 4 < 4.5

The profit made is:

3x + 2y = 3*2 + 2*4 = 6 + 8 = $14

For x = 3

y < (15 - 9)/2

y < 6/2

y < 3

The profit made is:

3x + 2y = 3*3 + 2*3 = 9 + 6 = $15

For x = 4:

y < (15 - 12)/2

y < 3/2

y < 1.5

Since y is integer, then y = 1 < 1.5

The profit made is:

3x + 2y = 3*4 + 2*1 = 12 + 2 = $14

**We notice that the maximum profit is $15.**

**We also notice that the baker can make maximum profit of $15, by baking 1 tray of corn muffins and 6 trays of bran muffins.**

**The baker can make maximum profit of $15, by baking 3 trays of corn muffins and 3 trays of bran muffins.**

The requirement of ingredients for a tray of corn muffin are 4 cups of milk and 3 cups of wheat powder. The requirements for a bran muffin are 2 cups of milk and 2 cups of wheat powder.

If the baker makes only corn muffins he can do 4 trays and profit he gets is $12 utilising all cups of milk and 12 cups of wheat powder.

If the baker bakes 1 tray of bran muffin, then from the remainig ingredientsof 14 milk cups and 13 cups of wheat powder, he can prepare 3 trays of corn muffin . So his profit is 1*2+3*3 = $11.

If he bakes 2 trays of bran muffins, then with the remaining ingredients (12milk cups and 11 wheat cups) he can prepare 3 trays of corn muffins. So his profit is 2*2+3*3 = $13.

If the baker bakes 3 trays of bran muffins, then he can bake from the remaining (10 m c and 9 w c) only 2 trays of corn muffin. The profit he gets is 3*3+2*3 = 12 dollar.

He can go for 4 tray of bran and 2 tray of corn muffins which require 16 cups of milk and 14 cups of wheat powder. This combination gets him 4*2+2*3 = $14.

Similarly baking 5 trays of bran muffins and 1 tray of wheat muffins requiring 14 cups of milk and 13 cups of wheat gives him 5*2+1*3 = 13 dollar.

If he chooses to bake 6 tray of bran and 1 tray of corn muffins, then he requires and utilises 16 cups of milk and 15 cups of wheat powder. The profit he gets is 6*2+3 = 15 dollars which is the maximum.

So he should do 6 trays of bran muffins and 1 tray of corn muffin and get the maximum profit of $15.

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