Maximizing Profit at a Plant Nursery

Question:

What is the optimal number of large oak trees and small oak trees that a plant nursery should have in its inventory each month to maximize profit? And what is the maximum profit that can be achieved?

Answer:

The nursery should have 280 large oak trees and 195 small oak trees in its inventory each month to maximize profit. The maximum profit that can be achieved is $16,075.

Explanation:

Linear Programming: To determine the optimal number of large and small oak trees that the nursery should have in its inventory each month to maximize profit, linear programming can be used. It is a mathematical technique that optimizes a linear objective function subject to linear constraints.

Objective Function: Let's define the decision variables: - x: Number of large oak trees in inventory - y: Number of small oak trees in inventory The profit for each large tree sold is $40, and the profit for each small tree sold is $25. Therefore, the objective function can be expressed as: Profit = 40x + 25y

Constraints: The constraints are: - The monthly demand is at most 475 oak trees: x + y ≤ 475 - The nursery does not want to allocate more than $43,000 each month on inventory for oak trees: 120x + 70y ≤ 43,000

The values of x and y need to be found that maximize the objective function while satisfying the constraints. This can be done through a graphical method or computer software.

Optimal Solution: The optimal solution is x = 280 (number of large oak trees) and y = 195 (number of small oak trees). The maximum profit that can be achieved is $16,075.

By having 280 large oak trees and 195 small oak trees in inventory each month, the plant nursery can maximize its profit and achieve the highest financial return.