Let's start by identifying our variables:
Set x as the number of rock CDs and y be the number of rap CDs.
Linear problems require defining the boundary conditions (constraints) that restrict our options, in this case: budget, production hours, release limits, and physical realities (you can't produce negative CDs).
Defining these boundaries first outlines our possible region, which represents all possible valid combinations of rock and rap CDs the company can produce.
Once the boundaries are set, we define the objective function, in this case total profit:
The fundamental theorem of linear programming states that the maximum/minimum value of a linear objective function over a convex feasible region always occurs at one of the vertices of that region. So, our strategy should be to identify the vertices formed by the intersecting constraint lines and evaluate the profit function at each corner point to find the highest payout.
To put this into a graphing calculator:
Budget Constraint: 15,000x + 12,000y ≤ 150,000 = 5x + 4y ≤ 50
Hours Constraint: 18x + 25y ≥ 175
Distributor Constraint: y ≤ x
Non-negativity Constraints: x ≥ 0, y ≥ 0
(Once you have the three vertices, use this.) Profit Function: P(x, y) = 20,000x + 30,000y
To see only the feasible region, use this equation. (Sometimes directly pasting this equation into Desmos doesn't work, so you may need to type this whole equation out. )
5x + 4y <= 50 {18x + 25y >= 175} {y <= x} {x >= 0} {y >= 0}
Solution:
The optimal production strategy to maximize profit is to produce 50/9 rock CDs (about 5.56 CDs) and 50/9 rap CDs (about 5.56 CDs), yielding a maximum profit of $277,777.78.
