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The problem states:

"To prevent pests, an orchard can have no more than 9 times as many apple trees as peach trees. Also, the number of apple trees plus 3 times the number of peach trees must not exceed 384. The revenue from a single apple tree is $88 and the revenue from a single peach tree is $106. Determine the number of each type of tree that will maximize revenue. What is the maximum revenue? Let **x** represent the number of apple trees and **y** represent the number of peach trees."

### Step-by-Step Solution:

We are given the following:
- **x** = number of apple trees.
- **y** = number of peach trees.

### Constraints:
1. The number of apple trees cannot be more than 9 times the number of peach trees:
   \[
   x \leq 9y
   \]
   
2. The number of apple trees plus 3 times the number of peach trees must not exceed 384:
   \[
   x + 3y \leq 384
   \]

### Objective:
We want to maximize the total revenue, which is given by:
\[
\text{Revenue} = 88x + 106y
\]

### Steps:

#### 1. Solve the constraints:
We have two inequalities:
- $ x \leq 9y $
- $ x + 3y \leq 384 $

From $ x \leq 9y $, we know that $ x $ must always be less than or equal to 9 times $ y $.

#### 2. Maximize the objective function:
We need to maximize $ 88x + 106y $ subject to the constraints.

Let’s first substitute $ x $ from $ x \leq 9y $ into the second constraint $ x + 3y \leq 384 $.

If $ x = 9y $, then:
\[
9y + 3y \leq 384
\]
\[
12y \leq 384
\]
\[
y \leq \frac{384}{12} = 32
\]

Thus, $ y \leq 32 $. The maximum number of peach trees is 32.

#### 3. Find $ x $ using the maximum $ y $:
If $ y = 32 $, then $ x = 9y = 9 \times 32 = 288 $.

#### 4. Calculate the maximum revenue:
Using $ x = 288 $ and $ y = 32 $, the total revenue is:
\[
\text{Revenue} = 88 \times 288 + 106 \times 32
\]
\[
\text{Revenue} = 25344 + 3392 = 28736
\]

### Final Answer:
- The number of apple trees, $ x $, is 288.
- The number of peach trees, $ y $, is 32.
- The maximum revenue is **$28,736**.

Would you like more details, or do you have any questions?

### 5 Related Questions:
1. How would the solution change if the maximum ratio of apple trees to peach trees was 8 instead of 9?
2. Can the same approach be applied to a different objective, such as minimizing costs instead of maximizing revenue?
3. What happens if the revenue from a single apple tree decreases to $80?
4. How would the problem be solved if the constraint on apple trees plus 3 times peach trees changed to a different equation?
5. Can you explain the difference between maximizing a linear function and optimizing a nonlinear one?

### Tip:
Always check if the constraints on a problem are linear, as this greatly simplifies finding the solution, often allowing you to use basic algebraic techniques or optimization methods like linear programming.

To prevent pests, an orchard can have no more than 9 times as many apple trees as peach trees. The number of apple trees plus 3 times the number of peach trees must not exceed 384. The revenue from a single apple tree is $88, and the revenue from a single peach tree is $106. What is the maximum revenue, and how many of each type of tree should be planted? Let x represent apple trees and y represent peach trees.

Maximize revenue for an orchard with constraints on the number of apple and peach trees. Learn how to apply linear programming concepts and solve for the maximum revenue when the number of apple trees cannot exceed 9 times the number of peach trees, and other constraints. Ideal for Grades 10-12.

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