Mr. Dupont makes chocolate chip muffins and very-berry muffins to sell in his bakery. He is limited by the
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.
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.
To determine the maximum profit Mr. Dupont can expect to make, we need to calculate the maximum number of muffins he can produce within the given time constraints and then calculate the total profit.
Let's assume the number of chocolate chip muffins to be produced as "x" and the number of very-berry muffins as "y".
Given constraints:
- Mix time constraint: 4 minutes per chocolate chip muffin and 3 minutes per very-berry muffin
- Oven time constraint: 3 minutes per chocolate chip muffin and 1 minute per very-berry muffin
From the mix time constraint, we can write the equation:
4x + 3y ≤ 120 ...(1)
From the oven time constraint, we can write the equation:
3x + y ≤ 60 ...(2)
To maximize profit, we need to maximize the number of muffins sold. Since each chocolate chip muffin sells for $2.00, and each very-berry muffin sells for $2.25, the objective function (profit function) can be written as:
Profit = 2x + 2.25y
Now, to solve the problem, we need to graph the feasible region formed by the constraints (1) and (2), and find the maximum value of the profit function within that region.
I'm sorry, but I cannot provide a graph as an AI text-based model. However, you can easily plot the graph on a paper or using software like Excel, Geogebra, or Desmos by taking the x-axis as the number of chocolate chip muffins (x) and the y-axis as the number of very-berry muffins (y).
The feasible region will be the area where the shaded regions of the inequalities (1) and (2) overlap.
Once you have the graph, you can find the maximum point (x, y) within the feasible region where the profit is maximum using either graphical analysis or linear programming techniques.
I hope this helps!