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Alan wants to bake blueberry muffins and bran muffins for the school bake sale. For a tray of blueberry muffins, he uses [tex]\( \frac{1}{3} \)[/tex] cup of oil and 2 eggs. For a tray of bran muffins, he uses [tex]\( \frac{1}{2} \)[/tex] cup of oil and 1 egg. Alan has 12 eggs on hand. He sells trays of blueberry muffins for [tex]$12 each and trays of bran muffins for $[/tex]10 each. To maximize the money raised at the bake sale, let [tex]\( x \)[/tex] represent the number of trays of blueberry muffins and [tex]\( y \)[/tex] the number of trays of bran muffins Alan bakes.

What are the constraints for the problem?

[tex]\[
\begin{array}{l}
\frac{1}{3} x + \frac{1}{2} y \leq 4 \\
2x + y \leq 12 \\
x \geq 0 \\
y \geq 0 \\
\end{array}
\][/tex]



Answer :

GinnyAnswer
Sure, let's outline the constraints given Alan's situation:

1. Alan uses [tex]\(\frac{1}{3}\)[/tex] cup of oil for each tray of blueberry muffins and [tex]\(\frac{1}{2}\)[/tex] cup of oil for each tray of bran muffins. Given that he has 4 cups of oil available, he needs to satisfy the constraint for oil usage:

[tex]\[ \frac{1}{3}x + \frac{1}{2}y \leq 4 \][/tex]

2. For eggs, Alan uses 2 eggs for each tray of blueberry muffins and 1 egg for each tray of bran muffins. Given that he has 12 eggs on hand, this results in the constraint for egg usage:

[tex]\[ 2x + y \leq 12 \][/tex]

3. Alan cannot bake a negative number of trays for either type of muffin, so the constraints for the number of trays baked are:

[tex]\[ x \geq 0 \][/tex]
[tex]\[ y \geq 0 \][/tex]

Combining these constraints, we have the following system of inequalities summarizing the problem:

[tex]\[ \begin{array}{l} \frac{1}{3}x + \frac{1}{2}y \leq 4 \\ 2x + y \leq 12 \\ x \geq 0 \\ y \geq 0 \\ \end{array} \][/tex]

Thus, these constraints must be satisfied in order to maximize the money raised at the bake sale.