Final answer:
To maximize profit, the dairy company should produce 300 units of cheese and 900 units of yogurt, as yogurt is more profitable and these quantities meet the minimum production requirements and the maximum capacity.
Step-by-step explanation:
To determine how many units of each type of dairy product the dairy company should make to maximize its profit, we need to solve a linear optimization problem. The company can produce a maximum of 1200 units of dairy products per week. It has to produce at least 300 units of cheese and 450 units of yogurt, given that the profit is $60 per unit of cheese and $90 per unit of yogurt.
Let's denote the number of cheese units by C and the number of yogurt units by Y. The objective is to maximize the profit function P = 60C + 90Y subject to the constraints:
- C ≥ 300 (at least 300 units of cheese)
- Y ≥ 450 (at least 450 units of yogurt)
- C + Y ≤ 1200 (cannot produce more than 1200 units in total)
To maximize profit, we must produce as many units of the more profitable item (yogurt) as possible while meeting the minimum requirements. Therefore, we would allocate the remaining production capacity after satisfying the minimum cheese requirement towards yogurt production.
Mathematically, since 300 units of cheese must be made, our profit maximizing combination would be to subtract those from the total 1200 units, leaving us with 900 units for yogurt (1200 - 300 = 900). Thus, the dairy should produce:
- 300 units of cheese (to satisfy the minimum requirement)
- 900 units of yogurt (to use the remaining capacity)
In conclusion, the dairy company should produce 300 units of cheese and 900 units of yogurt to maximize its profit.