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Step 1: Identify the unknowns and assign variables to them.
Let p represent the number of dozens of miniature cupcakes Hermione makes, and let c represent the number of dozens of miniature cakes Hermione makes.
Step 2: Write the objective function.
Hermione sells a dozen miniature cupcakes for $3.00 and a dozen miniature cakes for $5.00.
The objective function that represents the total profit she makes from selling these bakery items would be
P(p, c) = 3.00p + 5.00cP of p comma c equals three p plus 5 c
Step 3: Develop a system of inequalities that model the constraints identified in the problem.
The linear inequalities that model the number of miniature items Hermione should sell to maximize profits are:
Step 4: Graph the system of inequalities.
Note: The inequalities p ≥ 0p is greater than or equal to 0 and c ≥ 0c is greater than or equal to 0 are represented by Quadrant I and the x and y axes, respectively.
Step 5: Calculate the coordinates of the vertices of the shaded region formed by the system of equations.
The vertices of the commonly shaded region are (18, 0), (30, 0) and (2, 16).
Step 6: Find the vertex whose coordinates maximize or minimize the objective function.
We are trying to find how many miniature cupcakes and cakes Hermione should make to maximize profit while keeping within the limits previously set.
The coordinates of the vertices of the commonly shaded region and the objective function, P(p, c) = 3.00p + 5.00cP of p comma c equals 3 p plus 5 c, can help us find the answer to this question:
- P(18, 0) = 3.00(18) + 5.00(0) = 54p of 18 comma 0 equals 3 times 18 plus 5 times 0 equals 54
- P(30, 0) = 3.00(30) + 5.00(0) = 90p of 30 comma 0 equals 3 times 30 plus 5 times 0 equals 90
- P(2, 16) = 3.00(2) + 5.00(16) = 86p of 2 comma 16 equals 3 times 2 plus 5 times 16 equals 86
The vertex that returns the highest profit is (30, 0).
Step 7: Interpret the coordinates of the maximizing/minimizing vertex within the context of the problem.
Hermione should sell 30 miniature cupcakes and no miniature cakes to maximize her profits at the farmer’s market.
