Agriculture Your small farm encompasses 140 acres, and you are planning to grow tomatoes, lettuce, and carrots in the coming planting season. Fertilizer costs per acre are: \( \$ 5 \) for tomatoes, \( \$ 4 \) for lettuce, and \( \$ 2 \) for carrots. Based on past experience, you estimate that each acre of tomatoes will require an average of 4 hours of labor per week, while tending to lettuce and carrots will each require an average of 2 hours per week. You estimate a profit of \( \$ 2,000 \) for each acre of tomatoes, \( \$ 1,400 \) for each acre of lettuce and \( \$ 400 \) for each acre of carrots. You would like to spend at least \( \$ 560 \) on fertilizer (your niece owns the company that manufactures it) and your farm laborers can supply up to 700 hours per week. How many acres of each crop should you plant to maximize total profits? tomatoes lettuce carrots profit

To maximize profits while respecting your budget and labor constraints, consider this: your total fertilizer spending must be at least \$560. The costs per acre mean you’ll need to plant at least 9 acres of tomatoes (grab your calculator!). Each tomato acre costs \$5, leading to \$45, while lettuce and carrots require \$4 and \$2, respectively. Then, let’s tackle labor! With 4 hours for tomatoes and 2 hours for both lettuce and carrots per acre, keep an eye on that 700-hour limit. Now let's look at a practical approach: a linear programming model can help here! Set your variables for tomatoes (T), lettuce (L), and carrots (C). Now, establish your objective function as maximizing profit: \(2000T + 1400L + 400C\). Then incorporate your constraints based on fertilizer expenditure and labor hours. Graph this playfully, or use a software tool to find the optimal acres for each crop. Happy farming! 🌱