Maximizing Profit with Constraints?

📌 CONCEPT: A linear programming model is formulated to determine the optimal number of chocolate and vanilla cakes to produce for maximum profit, subject to constraints on production time.

📐 RULE / FORMULA: Let x be the number of chocolate cakes and y be the number of vanilla cakes. The objective function to maximize profit is P = 15x + 10y, subject to the constraints 2x + 1.5y ≤ 4 (production time constraint).

💡 WORKED EXAMPLE: Suppose the production time constraint is 2x + 1.5y ≤ 4. To maximize profit, we need to find the values of x and y that satisfy this constraint and give the maximum profit. Let's consider x = 1, then 2(1) + 1.5y ≤ 4, which gives 1.5y ≤ 2. Solving for y, we get y ≤ 4/1.5 = 8/3. This means we can produce at most 8/3 vanilla cakes when producing 1 chocolate cake.

⚠️ COMMON MISTAKE: Students often forget to include all constraints in the linear programming model or incorrectly assume the objective function as the only constraint.