3. Mr. Thibodeau 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.
• To meet demand Mr. Thibodeau must make at least
6 chocolate chip muffins and 9 very Berry muffins
• If Mr. Thibodeau 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. (8 marks)
To find the maximum profit Mr. Thibodeau can expect to make, we need to determine how many chocolate chip and very-berry muffins he should make to maximize his profit while still meeting the given constraints.
Let's define:
x = number of chocolate chip muffins
y = number of very-berry muffins
Based on the given constraints, we can create the following system of inequalities:
4x + 3y ≤ 120 (mixer availability constraint)
3x + y ≤ 60 (oven availability constraint)
x ≥ 6 (minimum number of chocolate chip muffins constraint)
y ≥ 9 (minimum number of very-berry muffins constraint)
We also need to consider the objective function, which is the profit. The profit for x chocolate chip muffins and y very-berry muffins can be calculated as:
Profit = (2x) + (2.25y)
To graph this problem, we'll graph the feasible region determined by the system of inequalities and then evaluate the profit function at each point in the feasible region. The maximum profit will occur at the point with the highest profit value.
Here is the graph of the feasible region:
Note: The feasible region is the shaded region below.
To find the maximum profit, we need to evaluate the profit function at each corner point of the feasible region.
Corner point A (0, 60):
Profit = (2(0)) + (2.25(60)) = 135
Corner point B (6, 60):
Profit = (2(6)) + (2.25(60)) = 135
Corner point C (18, 24):
Profit = (2(18)) + (2.25(24)) = 108
Corner point D (18, 9):
Profit = (2(18)) + (2.25(9)) = 63
Corner point E (6, 21):
Profit = (2(6)) + (2.25(21)) = 57.75
Therefore, the maximum profit Mr. Thibodeau can expect to make is $135 by making 0 chocolate chip muffins and 60 very-berry muffins.