To solve this problem, we seek to maximize the profit function given by [tex]\( P = 4x + 5y \)[/tex] subject to the constraints:
[tex]\[
\left\{
\begin{array}{l}
4x + 2y \leq 40 \\
-3x + y \geq -4 \\
x \geq 3 \\
y \geq 0
\end{array}
\right.
\][/tex]
Let's break down the steps to find the optimal values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that maximize [tex]\( P \)[/tex].
1. Identify the constraints and the objective function:
- Objective function: [tex]\( P = 4x + 5y \)[/tex]
- Constraints:
[tex]\[
\left\{
\begin{array}{l}
4x + 2y \leq 40 \\
-3x + y \geq -4 \\
x \geq 3 \\
y \geq 0
\end{array}
\right.
\][/tex]
2. Rewriting inequalities:
- The first constraint: [tex]\( 4x + 2y \leq 40 \)[/tex]
- The second constraint: [tex]\( -3x + y \geq -4 \)[/tex] (which can be rewritten as [tex]\( y \geq 3x - 4 \)[/tex])
- The third constraint: [tex]\( x \geq 3 \)[/tex]
- The fourth constraint: [tex]\( y \geq 0 \)[/tex]
3. Graphical method (optional for context):
To find the feasible region, we might graph these inequalities on the coordinate plane and identify the region where all these inequalities overlap. However, for complex inequalities, we typically employ linear programming techniques.
4. Finding the vertices of the feasible region:
The vertices of the feasible region are then calculated by solving the equations for each pair of constraints:
[tex]\[
\begin{align*}
4x + 2y &= 40 & y = 3x - 4 \\
4x + 2y &= 40 & x = 3 \\
3x - 4 &= x = 3 & y = 0 \text { and } x = 3 \\
\end{align*}
\][/tex]
From solving these pairs and testing the points within the constraints, we can identify the feasible region's vertices. Given the solution, the optimal values are derived from this feasible region.
5. Maximizing the objective function:
By evaluating [tex]\( P = 4x + 5y \)[/tex] at the critical points, we find the value of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that maximize [tex]\( P \)[/tex].
Based on the obtained solutions:
[tex]\[
(x, y) = (4.8, 10.4)
\][/tex]
we substitute these values back into the profit function:
[tex]\[
P = 4(4.8) + 5(10.4)
\][/tex]
[tex]\[
P = 19.2 + 52
\][/tex]
[tex]\[
P = 71.2
\][/tex]
Thus, the maximum profit is:
[tex]\[
P_{\text{max}} = 71.20 \text{ (rounded to the nearest cent)}
\][/tex]
Therefore, the optimal solution (rounded to the nearest cent) is:
[tex]\[
x = 4.8, \quad y = 10.4, \quad P_{\text{max}} = 71.20
\][/tex]
