To determine the constraints for Alan's problem of maximizing the money raised by baking blueberry muffins and bran muffins, we need to account for the resources available and the minimum requirements. Let's break it down step-by-step:
1. Oil constraint:
- Each tray of blueberry muffins uses [tex]\(\frac{1}{3}\)[/tex] cup of oil.
- Each tray of bran muffins uses [tex]\(\frac{1}{2}\)[/tex] cup of oil.
- Alan has a total of 4 cups of oil.
Therefore, the constraint for the amount of oil can be represented as:
[tex]\[
\frac{1}{3}x + \frac{1}{2}y \leq 4
\][/tex]
2. Egg constraint:
- Each tray of blueberry muffins uses 2 eggs.
- Each tray of bran muffins uses 1 egg.
- Alan has a total of 12 eggs.
Therefore, the constraint for the number of eggs can be represented as:
[tex]\[
2x + y \leq 12
\][/tex]
3. Non-negativity constraints:
- Alan cannot bake a negative number of trays, so both [tex]\(x\)[/tex] and [tex]\(y\)[/tex] must be non-negative.
These constraints are:
[tex]\[
x \geq 0
\][/tex]
[tex]\[
y \geq 0
\][/tex]
To summarize, the constraints for this linear programming problem are:
[tex]\[
\begin{aligned}
&\frac{1}{3}x + \frac{1}{2}y \leq 4 \\
&2x + y \leq 12 \\
&x \geq 0 \\
&y \geq 0
\end{aligned}
\][/tex]
