To find the maximum of the profit function [tex]\( P = 2x + 10y \)[/tex] subject to the given constraints, we need to analyze the feasible region determined by the constraints:
1. [tex]\( 4x + 2y \leq 5 \)[/tex]
2. [tex]\( -3x + y \geq -2 \)[/tex]
3. [tex]\( x \geq 0 \)[/tex]
4. [tex]\( y \geq 0 \)[/tex]
We will identify the boundary lines of the constraints and determine the feasible region. This is where the solution to our optimal problem lies.
### Step 1: Identify the boundary lines of the constraints
1. For the constraint [tex]\( 4x + 2y \leq 5 \)[/tex]:
- The equation of the line is [tex]\( 4x + 2y = 5 \)[/tex].
- Solving for [tex]\( y \)[/tex], we get [tex]\( y = \frac{5 - 4x}{2} \)[/tex].
2. For the constraint [tex]\( -3x + y \geq -2 \)[/tex]:
- The equation of the line is [tex]\( -3x + y = -2 \)[/tex].
- Solving for [tex]\( y \)[/tex], we get [tex]\( y = 3x - 2 \)[/tex].
3. For the constraints [tex]\( x \geq 0 \)[/tex] and [tex]\( y \geq 0 \)[/tex], these are simply the axes [tex]\( x = 0 \)[/tex] and [tex]\( y = 0 \)[/tex], respectively.
### Step 2: Find the intersection points of the boundary lines
To identify the vertices of the feasible region, we find the intersection points of the lines described by the constraints:
1. Intersection of [tex]\( 4x + 2y = 5 \)[/tex] and [tex]\( -3x + y = -2 \)[/tex]:
- Solve these two equations simultaneously.
[tex]\[
4x + 2(3x - 2) = 5
\][/tex]
[tex]\[
4x + 6x - 4 = 5
\][/tex]
[tex]\[
10x - 4 = 5
\][/tex]
[tex]\[
10x = 9 \implies x = \frac{9}{10} = 0.9
\][/tex]
Substituting [tex]\( x = 0.9 \)[/tex] back into [tex]\( y = 3x - 2 \)[/tex]:
[tex]\[
y = 3(0.9) - 2 = 2.7 - 2 = 0.7
\][/tex]
Therefore, one point is [tex]\( \left(0.9, 0.7\right) \)[/tex].
### Step 3: Check intersections with the axes
- When [tex]\( x = 0 \)[/tex] in [tex]\( 4x + 2y = 5 \)[/tex]:
[tex]\[
2y = 5 \implies y = \frac{5}{2} = 2.5
\][/tex]
Intersection is [tex]\( (0, 2.5) \)[/tex].
- When [tex]\( y = 0 \)[/tex] in [tex]\( 4x + 2y = 5 \)[/tex]:
[tex]\[
4x = 5 \implies x = \frac{5}{4} = 1.25
\][/tex]
Intersection is [tex]\( (1.25, 0) \)[/tex].
### Step 4: Evaluate the profit function at these points
The feasible vertices are:
- [tex]\( (0, 2.5) \)[/tex]
- [tex]\( (0.9, 0.7) \)[/tex]
- [tex]\( (1.25, 0) \)[/tex]
- (plus origin which generally does not maximize the profit function)
Evaluate at each point:
- For [tex]\( (0, 2.5) \)[/tex]: [tex]\( P = 2(0) + 10(2.5) = 0 + 25 = 25 \)[/tex]
- For [tex]\( (1.25, 0) \)[/tex]: [tex]\( P = 2(1.25) + 10(0) = 2.5 + 0 = 2.5 \)[/tex]
- For [tex]\( (0.9, 0.7) \)[/tex]: [tex]\( P = 2(0.9) + 10(0.7) = 1.8 + 7 = 8.8 \)[/tex]
### Step 5: Identify the maximum profit
From the calculated profits:
- [tex]\( (0, 2.5) \)[/tex] yields a profit of [tex]\( 25 \)[/tex]
- [tex]\( (0.9, 0.7) \)[/tex] yields a profit of [tex]\( 8.8 \)[/tex]
- [tex]\( (1.25, 0) \)[/tex] yields a profit of [tex]\( 2.5 \)[/tex]
The maximum profit is [tex]\( 25 \)[/tex] at the point [tex]\( (0, 2.5) \)[/tex].
Thus, the maximum profit is:
[tex]\[
\boxed{(0, 2.50, 25.00)}
\][/tex]
