can anyone please explain to me how you do this word problem?

baking a tray of corn muffins takes 4 c milk and 3 c wheat flour. A tray of bran muffins takes 2 c of milk and 3 c of flour. A baker has 16 c milk and 15 c wheat flower. He makes 3$profit per tray of corn muffins and 2$ profit of bran muffins. How many trays of each type of muffin should the baker make to maximize his profit?

2. Hello, freakyfrog!

Are you skimming through some book?
This is a Linear Programming problem.
If this is an assignment, you should have been taught the procedure.

A tray of corn muffins takes 4 c milk and 3 c wheat flour.
A tray of bran muffins takes 2 c of milk and 3 c of flour.
A baker has 16 c milk and 15 c wheat flower.
He makes $3 profit per tray of corn muffins and$2 profit of bran muffins.
How many trays of each type of muffin should the baker make to maximize his profit?

Let $\displaystyle x$ = number of trays of corn muffins, $\displaystyle x \geq 0$
Let $\displaystyle y$ = number of trays of bran muffins, $\displaystyle y \geq 0$

Place the information in a chart:

. . $\displaystyle \begin{array}{c||c|c||c} & \text{milk} & \text{flour} & \text{Profit} \\ \hline \text{corn }(x) & 4x & 3x & 3x \\ \text{bran }(y) & 2y & 3y & 2y \\ \hline \text{Total} & 16 & 15 \end{array}$

We have: .$\displaystyle \begin{array}{cccccccc}4x + 2y & \leq & 16 & \Rightarrow & 2x + y & \leq & 8 & [1] \\ 3x + 3y & \leq & 15 & \Rightarrow & x + y &\leq & 5 & [2] \end{array}$

[1] Graph the line: $\displaystyle 2x + y \:=\:8$
. . Intercepts: (4,0), (0,8)
. . Shade the region below the line.

[2] Graph the line: $\displaystyle x + y \:=\:5$
. . Intercepts: (5,0), (0,5)
. . Shade the region below the line.

The shaded region looks like this:
Code:
      |
8 *
|*
| *
5 o  *
|:* *
|:::**
|:::::o
|::::::**
|::::::::* *
|::::::::*  *
- - o - - - - o - * - -
4   5

We are concerned with only the vertices of the shaded region.

We see three of them: (0,0), (4,0), (0,5)

The fourth is the intersection of the two lines.
Solve the system and we get: (3,2)

Test these vertices in the profit function: .$\displaystyle P \:=\:3x+2y$
. . and see which one produces maximum profit.

. . $\displaystyle \begin{array}{c|c} & \text{Profit} \\ \text{Vertex} & 3x+2y \\ \hline (0,0) & 0 \\ (4,0) & 12 \\ (0,5) & 10 \\ {\color{blue}(3,2)} & {\color{blue}13} \end{array}$

For maximum profit, he should make 3 trays of corn muffins and 2 trays of bran muffins.

3. Originally Posted by freakyfrog
can anyone please explain to me how you do this word problem?

baking a tray of corn muffins takes 4 c milk and 3 c wheat flour. A tray of bran muffins takes 2 c of milk and 3 c of flour. A baker has 16 c milk and 15 c wheat flower. He makes 3$profit per tray of corn muffins and 2$ profit of bran muffins. How many trays of each type of muffin should the baker make to maximize his profit?
This is a linear programming model. We need to set up a set of constraints, graph them and locate the vertices of the feasible region to find the right combination to maximize profit.

Let x = no. of corn muffins to make
Let y = no. of bran muffins to make

(1) $\displaystyle x\ge 0$
(2) $\displaystyle y\ge 0$
(3) $\displaystyle 4x + 2y \leq 16\Longrightarrow y\leq8-2x$
(4) $\displaystyle 3x+3y \leq 15\Longrightarrow y\leq5-x$

Profit Function=$\displaystyle P(x,y)=3x+2y$

I graphed the inequalities on a TI-84+ and isolated the feasible region. The vertices are (4, 0), (0, 0), (0, 5), (3, 2)

Substitute these points into your profit function and see which set makes P(x, y) the largest.

I don't know what tools you are allowed to use. If you had to do this without a calculator, you would have to find the intersections of the lines using your favorite method of solving simultaneous linear equations.

Good luck.

EDIT: Soroban, even with all that code you write, I'm too slow.