To solve this problem, let’s define our variables and constraints based on the given conditions.

Given Information

1. Total Land: The farmer has 30 acres in total to divide between crop A and crop B.
2. Cost of Seed:
   • Crop A: $10 per acre
   • Crop B: $20 per acre
3. Maximum Budget: The farmer can spend at most $400 on seeds.
4. Profit per Acre:
   • Crop A: $160 per acre
   • Crop B: $130 per acre

Variables

Let:

• $x$ = acres planted with crop A
• $y$ = acres planted with crop B

Constraints

1. Land Constraint: $x + y = 30$

2. Budget Constraint: $10x + 20y \leq 400$

3. Objective Function (Profit): We want to maximize the total profit $P$, which is: $P = 160x + 130y$

Steps to Solve

1. Substitute $y = 30 - x$ (from the land constraint) into the budget constraint and the objective function.
2. Solve the budget constraint: Substitute $y = 30 - x$ into $10x + 20y \leq 400$: $10x + 20(30 - x) \leq 400$ Simplify this inequality to find the feasible values of $x$.
3. Maximize the profit function by substituting feasible values of $x$ and corresponding values of $y$ into $P = 160x + 130y$.

Let's go through these calculations:The optimal solution is:

• Acres of crop A: 20 acres
• Acres of crop B: 10 acres
• Maximum Profit: $4,500

By planting 20 acres of crop A and 10 acres of crop B, the farmer maximizes her profit to $4,500 while staying within the budget and land constraints.

Would you like a further breakdown of these calculations or have any questions?

Here are some related questions to deepen understanding:

1. How would the solution change if the farmer had only $300 to spend on seed?
2. What if crop A and crop B both required the same amount of seed cost per acre?
3. How would different profit values for each crop affect the optimal planting strategy?
4. Can this type of problem be solved using graphical methods, and how?
5. How would you set up this problem if there were three crops instead of two?

Tip: Always double-check constraints to ensure they are correctly applied in optimization problems.