https://file.aiquickdraw.com/mathbot/file/1730050168936cbubqlof.jpg

Let's go through this optimization problem step-by-step.

### Problem Summary
A farmer has:
1. **30 acres** to divide between crops A and B.
2. **Budget constraint**: Can spend **at most $400** on seed.
3. **Costs**:
   - Crop A seed: **$10 per acre**
   - Crop B seed: **$20 per acre**
4. **Profits**:
   - Crop A: **$160 per acre**
   - Crop B: **$130 per acre**

The goal is to determine how many acres should be dedicated to each crop to maximize profit, subject to the constraints.

### Step 1: Define Variables
Let:
- $ x $ = acres of crop A
- $ y $ = acres of crop B

### Step 2: Set Up Constraints
1. **Acreage Constraint**: The total acres for both crops cannot exceed 30.
   \[
   x + y = 30
   \]

2. **Budget Constraint**: The total cost for seeds cannot exceed $400.
   \[
   10x + 20y \leq 400
   \]

### Step 3: Write the Profit Function
The profit for each crop is:
- Crop A: $ 160x $
- Crop B: $ 130y $

So, the total profit function $ P $ to maximize is:
\[
P = 160x + 130y
\]

### Step 4: Simplify the Constraints
From the acreage constraint $ x + y = 30 $, we can express $ y $ in terms of $ x $:
\[
y = 30 - x
\]

Substitute $ y = 30 - x $ into the budget constraint:
\[
10x + 20(30 - x) \leq 400
\]
Expanding and simplifying:
\[
10x + 600 - 20x \leq 400
\]
\[
-10x \leq -200
\]
\[
x \geq 20
\]

This implies:
- $ x $ (acres of crop A) can be between **20 and 30 acres**.
- $ y = 30 - x $, so $ y $ (acres of crop B) will be between **0 and 10 acres**.

### Step 5: Substitute and Maximize the Profit
We need to evaluate $ P = 160x + 130y $ for the feasible values of $ x $ and $ y $.

1. **If $ x = 20 $:**
   - $ y = 30 - 20 = 10 $
   - Profit $ P = 160(20) + 130(10) = 3200 + 1300 = 4500 $

2. **If $ x = 30 $:**
   - $ y = 30 - 30 = 0 $
   - Profit $ P = 160(30) + 130(0) = 4800 $

### Conclusion
The maximum profit is achieved when:
- **30 acres are planted with crop A**, and
- **0 acres are planted with crop B**.

This setup yields a maximum profit of **$4800**.

Would you like further details or have any questions?

---

### Related Practice Questions
1. How would the result change if crop B’s seed cost decreased to $15 per acre?
2. What if the profit per acre for crop A increased to $200?
3. How would the optimal planting strategy change with a 40-acre farm instead of 30 acres?
4. What if the farmer’s budget increased to $600 for seeds?
5. How could you set up a linear programming problem to solve similar optimization issues?

### Tip
Always start by defining variables and constraints clearly in optimization problems; it helps streamline the entire solution process.

A farmer is going to divide her 30-acre farm between two crops. Seed for crop A costs $10 per acre. Seed for crop B costs $20 per acre. The farmer can spend at most $400 on seed. If crop B brings in a profit of $130 per acre, and crop A brings in a profit of $160 per acre, how many acres of each crop should the farmer plant to maximize her profit?

Learn how to maximize profit by optimally distributing 30 acres between two crops. This problem covers linear programming and algebraic optimization, with real-world constraints like budget limits and profitability per acre. Ideal for high school students in Grades 10-12.

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