Maximizing Profit by Picking and Packing Apples and Pears

Q1. How can we determine the number of pecks of apples and pears that should be picked and packed every hour to maximize profit?

Solving the Linear Program Graphically

To maximize profit by picking and packing apples and pears, we can set up a linear programming model. Let x represent the number of pecks of apples and y represent the number of pecks of pears picked and packed every hour. The objective is to maximize profit, given as 4x + 3y. The constraints include total picking and packing time not exceeding 60 minutes and the non-negativity of the number of pecks. The time constraints for picking and packing apples and pears lead to the inequalities: - For picking apples: 4x + 5y ≤ 60 - For packing apples: 4x + 4y ≤ 60 The non-negativity constraints are: - x ≥ 0 - y ≥ 0 By solving graphically or using a calculator, we find that picking and packing 12 pecks of apples and 6 pecks of pears every hour would result in a profit of $66. This optimal solution is obtained at x = 12 and y = 6. This indicates that 12 pecks of apples and 6 pecks of pears should be picked and packed every hour to maximize profit.

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