To find the maximum for the profit function [tex]\( P = 2x + 10y \)[/tex] subject to the given constraints, we need to follow a systematic approach, including formulating the problem as a linear programming (LP) problem, identifying the constraints, setting up the feasible region, and then finding the solutions at the boundary points of the feasible region. Below are the detailed steps:
### Step-by-Step Solution:
1. Rewrite the constraints to standard form for linear programming:
[tex]\[
\begin{cases}
3x - 3y \geq -10 & \implies -3x + 3y \leq 10 \\
6x - 2y \leq 5 \\
x \geq 0 \\
y \geq 0 \\
\end{cases}
\][/tex]
2. Identify the linear inequalities and define them properly:
- Constraint 1: [tex]\( -3x + 3y \leq 10 \)[/tex]
- Constraint 2: [tex]\( 6x - 2y \leq 5 \)[/tex]
- Constraint 3: [tex]\( x \geq 0 \)[/tex]
- Constraint 4: [tex]\( y \geq 0 \)[/tex]
3. Formulate the coefficients for the objective function and constraints:
- Objective Function: Maximize [tex]\( P = 2x + 10y \)[/tex]
- Coefficients for the inequalities:
- For the first constraint: [tex]\([-3, 3]\)[/tex]
- For the second constraint: [tex]\([6, -2]\)[/tex]
- Right-hand side of the constraints: [tex]\([10, 5]\)[/tex]
- Bounds for the variables: [tex]\( x \geq 0 \)[/tex] and [tex]\( y \geq 0 \)[/tex]
4. Set up the linear programming problem to maximize the profit function.
5. Find the feasible region by solving the inequalities:
- Plot the lines defined by the equations converted from inequalities:
- Line 1: [tex]\( -3x + 3y = 10 \)[/tex]
- Line 2: [tex]\( 6x - 2y = 5 \)[/tex]
- Check the intersection points and boundary conditions to identify the feasible region bounded by values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
6. Evaluate the objective function at the vertices of the feasible region to determine the maximum value:
- Identify the intersection points of the constraints and boundaries.
- These are the points where the constraints are binding.
8. Determine values at intersection points and boundaries:
Using these steps, we reach the feasible solution and compute the optimal values [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that will maximize the profit function [tex]\( P = 2x + 10y \)[/tex].
### Solution:
Based on the feasible region and evaluating the objective function at the critical points determined from the constraints, the optimal solution is:
- Optimal values for [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
x \approx 2.92, \quad y \approx 6.25
\][/tex]
- Maximum Profit:
[tex]\[
P \approx 68.33
\][/tex]
So, the optimal point is [tex]\( (x, y) = (2.92, 6.25) \)[/tex], and the maximum profit is [tex]\( P = 68.33 \)[/tex].
Therefore, the values rounded to the nearest cent are:
- [tex]\( x = 2.92 \)[/tex]
- [tex]\( y = 6.25 \)[/tex]
- Maximum Profit [tex]\( P = 68.33 \)[/tex]
These values ensure that the profit function [tex]\( P = 2x + 10y \)[/tex] is maximized while satisfying all the given constraints.
