Final answer:
To maximize their profit within their budget constraints on selling shirts and hats, the junior class needs to solve a set of inequalities using linear programming. The profit is calculated based on the cost and selling price of the shirts and hats, and constraints include their budget, demand for more hats than shirts, and hats' supply limit.
Step-by-step explanation:
The student's question revolves around linear programming and optimization of profit given a set of constraints on budget and quantities. To determine the maximum profit that the junior class at a certain school can earn from selling shirts and hats within their budget of $875, we need to establish variables for the number of shirts (s) and hats (h) they will purchase and sell. They can spend $3.50 per shirt and $1.25 per hat, and they need to purchase at least 150 more hats than shirts while the maximum number of hats available is 420.
First, set up the constraint equations based on the given information:
- 3.50s + 1.25h ≤ 875 (Budget constraint)
- h ≥ s + 150 (At least 150 more hats than shirts)
- h ≤ 420 (Maximum number of hats available)
Next, formulate the profit function they wish to maximize. Profit is calculated by the selling price minus the cost price for each item:
- Profit from shirts = (5 - 3.50)s = 1.50s
- Profit from hats = (2 - 1.25)h = 0.75h
- Total profit = 1.50s + 0.75h
The class should use graphical methods or linear programming techniques to solve these inequalities and maximize their total profit. Since exact figures are not asked for, we don’t need to provide numerical solutions.