Question

A farmer, Caitlin, has one high-yielding field and one low-yielding field. On the good field, she can produce 100 units of corn for $1.00 each, a second $100 for $1.20 each, and the third hundred at $1.40 each, etc. In the bad field, it’s $2.00 each for the first hundred, $2.50 for the 2nd, $3.00 for the 3rd hundred, etc. Corn can be sold at $3.15. How many units of corn should she grow? What will her profits be?

Answer 1

Farmer Caitlin should grow 1100 units of corn in the high-yielding field to maximize her profits. Any further production would result in marginal costs exceeding the sale price. Her total profit from the 1100 units would be $1245.

Step-by-step explanation:

The question involves determining the optimal number of corn units that farmer Caitlin should grow to maximize her profits, given the cost of production and the sale price. To solve this, we calculate the profit for each additional hundred units of corn in the high-yielding and low-yielding fields until the cost is equal to or greater than the sale price of $3.15.

In the high-yielding field, the profit margins decrease by $0.20 for each additional hundred units ($1.00, $1.20, $1.40, etc.), while in the low-yielding field, they increase by $0.50 ($2.00, $2.50, $3.00, etc.). Caitlin should continue to produce corn until her marginal cost is less than or equal to the sale price of $3.15.

1. 100 units at $1.00 = $2.15 profit/unit
2. 100 units at $1.20 = $1.95 profit/unit
3. 100 units at $1.40 = $1.75 profit/unit
4. 100 units at $1.60 = $1.55 profit/unit
5. 100 units at $1.80 = $1.35 profit/unit
6. 100 units at $2.00 = $1.15 profit/unit
7. 100 units at $2.20 = $0.95 profit/unit
8. 100 units at $2.40 = $0.75 profit/unit
9. 100 units at $2.60 = $0.55 profit/unit
10. 100 units at $2.80 = $0.35 profit/unit

At this point, Caitlin should stop growing corn in the high-yielding field since the next hundred would cost $3.00 to produce, which is below the selling price. In the low-yielding field, she only produces at a loss, as the costs are already above the selling price. Therefore, Caitlin should grow 1100 units of corn in the high-yielding field for a total profit of $2.15x100 + $1.95x100 + ... + $0.35x100.

To find the exact total profit, we would sum these individual profits: $2.15(100) + $1.95(100) + $1.75(100) + $1.55(100) + $1.35(100) + $1.15(100) + $0.95(100) + $0.75(100) + $0.55(100) + $0.35(100) = $215 + $195 + $175 + $155 + $135 + $115 + $95 + $75 + $55 + $35 = $1245 total profit.