Mr. Dupont makes chocolate chip muffins and very-berry muffins to sell in his bakery. He is limited by the

Question

following constraints:
• Each chocolate chip muffin requires 4 minutes mix time and the very-berry muffins require 3
minutes mix time. The mixer is only available for 2 hours (120 minutes) each day.
• Each chocolate chip muffin takes 3 minutes bake time and the very-berry muffins take 1 minute
bake time. The oven is only available for 1 hour (60 min) each day.
If Mr. Dupont sells the chocolate chip muffins for $2.00 each and the very-berry muffins for $2.25 each, what
is the maximum profit he can expect to make? Include your graph with the solution.

Answer 1

To solve this problem, we need to determine how many chocolate chip muffins and very-berry muffins Mr. Dupont should make in order to maximize profit while staying within the time constraints.

Let's define:
x = number of chocolate chip muffins
y = number of very-berry muffins

We want to maximize the profit function:
P = 2x + 2.25y

Subject to the following constraints:
4x + 3y ≤ 120 (mixing time constraint)
3x + y ≤ 60 (baking time constraint)

We can plot these constraints on a graph:

First constraint: 4x + 3y ≤ 120
If we set x = 0, we have 3y ≤ 120. Solving for y, we get y ≤ 40.
If we set y = 0, we have 4x ≤ 120. Solving for x, we get x ≤ 30.
Plotting these limits on a graph, we get a line.

Second constraint: 3x + y ≤ 60
If we set x = 0, we have y ≤ 60. Plotting this limit on the same graph, we get another line.
If we set y = 0, we have 3x ≤ 60. Plotting this limit on the same graph, we get another line.

The feasible region where the two constraints intersect represents the area where both constraints are satisfied.

Calculating the vertices of this feasible region:
- Let x = 0, then y = 40.
- Let y = 0, then x = 20.
- Let 3x + y = 60 and solve the system of equations with 4x + 3y = 120. We get x = 24 and y = 12.

Now we can substitute the values of x and y into the profit function to find the maximum profit:
P = 2(0) + 2.25(40) = $90 (Point A)
P = 2(20) + 2.25(0) = $40 (Point B)
P = 2(24) + 2.25(12) = $105 (Point C)

The maximum profit is $105.