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.