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what is the maximum value of the function

We are tasked with maximizing the objective function \(3x + 5y\) under the following constraints:

\[
\begin{aligned}
    x + 2y &\leq 32 \\
    3x + 2y &\geq 36 \\
    x &\leq 4 \\
    x &\geq 0, \quad y \geq 0
\end{aligned}
\]

This is a linear programming problem. The first step is to determine the feasible region by plotting the constraints and identifying the points where they intersect, then evaluating the objective function at those points.

### Step 1: Solve for boundary points.
Let's find the intersection points of the constraints.

#### 1. Solve for the intersections between the constraints.

1. \( x + 2y = 32 \)
2. \( 3x + 2y = 36 \)

Subtracting the first equation from the second:

\[
(3x + 2y) - (x + 2y) = 36 - 32 \\
2x = 4 \\
x = 2
\]

Substitute \(x = 2\) into the first equation to find \(y\):

\[
2 + 2y = 32 \\
2y = 30 \\
y = 15
\]

So, one intersection point is \( (2, 15) \).

#### 2. Solve other constraints by checking the boundaries:
Using \(x = 4\) and \(x = 0\) with the constraints.

#### Feasible points:
- \( (2, 15) \)
- \( (4, 0) \)
- Other boundary points.

### Step 2: Evaluate the objective function \(3x + 5y\) at these points.

1. At \( (2, 15) \): 
\[
3(2) + 5(15) = 6 + 75 = 81
\]

2. At \( (4, 0) \):
\[
3(4) + 5(0) = 12
\]

### Conclusion:
The maximum value of the objective function is 81.

Would you like further details or have any questions? Here are five related questions you may find helpful:

1. How do we graphically interpret the feasible region in linear programming?
2. What is the difference between maximizing and minimizing objective functions?
3. How do corner points help in finding optimal solutions in linear programming?
4. How can the Simplex Method be applied to this problem?
5. What happens if the constraints do not form a bounded region?

**Tip:** Always check all corner points of the feasible region, as the maximum (or minimum) of the objective function occurs at one of those points.

Maximize the objective function 3x + 5y subject to the constraints: x + 2y ≤ 32, 3x + 2y ≥ 36, x ≤ 4, x ≥ 0, y ≥ 0.

Learn how to solve a linear programming problem by maximizing the objective function 3x + 5y under given constraints: x + 2y ≤ 32, 3x + 2y ≥ 36, x ≤ 4, and non-negative x, y. This problem involves key concepts such as systems of inequalities and the corner point theorem, suitable for students in Grades 11-12.

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