Answer :
To find the maximum value of [tex]\( P = 9x + 8y \)[/tex] subject to the given constraints, we need to perform the following steps:
### Step 1: Identify and Graph the Constraints
First, let's convert each inequality constraint into an equation to identify the boundary lines and their intercepts:
1. For the inequality [tex]\( 8x + 6y \leq 48 \)[/tex]:
- The equation form is [tex]\( 8x + 6y = 48 \)[/tex].
- To find the x-intercept, set [tex]\( y = 0 \)[/tex]:
[tex]\[ 8x = 48 \implies x = \frac{48}{8} = 6 \][/tex]
- Therefore, the x-intercept of this constraint is [tex]\( x = 6 \)[/tex].
2. For the inequality [tex]\( 7x + 7y \leq 49 \)[/tex]:
- The equation form is [tex]\( 7x + 7y = 49 \)[/tex].
- To find the x-intercept, set [tex]\( y = 0 \)[/tex]:
[tex]\[ 7x = 49 \implies x = \frac{49}{7} = 7 \][/tex]
- Therefore, the x-intercept of this constraint is [tex]\( x = 7 \)[/tex].
3. Additionally, we have the non-negativity constraints [tex]\( x \geq 0 \)[/tex] and [tex]\( y \geq 0 \)[/tex], which means that [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are restricted to the first quadrant.
### Step 2: Determine the Feasible Region
The feasible region is the area where all the constraints overlap on the graph:
1. Draw the line [tex]\( 8x + 6y = 48 \)[/tex]. This line passes through [tex]\( (6, 0) \)[/tex] (x-intercept) and [tex]\( (0, 8) \)[/tex] (y-intercept).
2. Draw the line [tex]\( 7x + 7y = 49 \)[/tex]. This line passes through [tex]\( (7, 0) \)[/tex] (x-intercept) and [tex]\( (0, 7) \)[/tex] (y-intercept).
3. Shade the region that satisfies both inequalities.
### Step 3: Identify the Corner Points
Determine the intersection points of the boundary lines within the feasible region. The feasible region for linear programming problems is typically a polygon, and the maximum or minimum values occur at the vertices (corner points) of this region.
To find the intersection of [tex]\( 8x + 6y = 48 \)[/tex] and [tex]\( 7x + 7y = 49 \)[/tex], solve the system of equations:
- [tex]\( 8x + 6y = 48 \)[/tex]
- [tex]\( 7x + 7y = 49 \)[/tex]
Multiply the second equation by [tex]\(\frac{6}{7}\)[/tex] to match the coefficients for [tex]\( y \)[/tex]:
[tex]\[ 6x + 6y = 42 \quad (scaled second equation) \][/tex]
Subtract this from the first equation:
[tex]\[ (8x + 6y) - (6x + 6y) = 48 - 42 \][/tex]
[tex]\[ 2x = 6 \implies x = 3 \][/tex]
Substitute [tex]\( x = 3 \)[/tex] back into the scaled second equation:
[tex]\[ 6(3) + 6y = 42 \implies 18 + 6y = 42 \implies 6y = 24 \implies y = 4 \][/tex]
Hence, the intersection point is [tex]\( (3, 4) \)[/tex].
### Step 4: Evaluate the Objective Function at Each Vertex
Evaluate [tex]\( P = 9x + 8y \)[/tex] at each vertex of the feasible region:
1. At [tex]\( (0, 0) \)[/tex]:
[tex]\[ P = 9(0) + 8(0) = 0 \][/tex]
2. At [tex]\( (6, 0) \)[/tex]:
[tex]\[ P = 9(6) + 8(0) = 54 \][/tex]
3. At [tex]\( (7, 0) \)[/tex]:
[tex]\[ P = 9(7) + 8(0) = 63 \][/tex]
4. At the intersection point [tex]\( (3, 4) \)[/tex]:
[tex]\[ P = 9(3) + 8(4) = 27 + 32 = 59 \][/tex]
### Step 5: Identify the Maximum Value
Based on these evaluations, the maximum value of [tex]\( P = 9x + 8y \)[/tex] within the feasible region is [tex]\( 59 \)[/tex], which occurs at the point [tex]\( (3, 4) \)[/tex].
Thus, the maximum value of [tex]\( P \)[/tex] is [tex]\(\boxed{59}\)[/tex], with the optimal point being [tex]\( x = 3 \)[/tex] and [tex]\( y = 4 \)[/tex].
### Step 1: Identify and Graph the Constraints
First, let's convert each inequality constraint into an equation to identify the boundary lines and their intercepts:
1. For the inequality [tex]\( 8x + 6y \leq 48 \)[/tex]:
- The equation form is [tex]\( 8x + 6y = 48 \)[/tex].
- To find the x-intercept, set [tex]\( y = 0 \)[/tex]:
[tex]\[ 8x = 48 \implies x = \frac{48}{8} = 6 \][/tex]
- Therefore, the x-intercept of this constraint is [tex]\( x = 6 \)[/tex].
2. For the inequality [tex]\( 7x + 7y \leq 49 \)[/tex]:
- The equation form is [tex]\( 7x + 7y = 49 \)[/tex].
- To find the x-intercept, set [tex]\( y = 0 \)[/tex]:
[tex]\[ 7x = 49 \implies x = \frac{49}{7} = 7 \][/tex]
- Therefore, the x-intercept of this constraint is [tex]\( x = 7 \)[/tex].
3. Additionally, we have the non-negativity constraints [tex]\( x \geq 0 \)[/tex] and [tex]\( y \geq 0 \)[/tex], which means that [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are restricted to the first quadrant.
### Step 2: Determine the Feasible Region
The feasible region is the area where all the constraints overlap on the graph:
1. Draw the line [tex]\( 8x + 6y = 48 \)[/tex]. This line passes through [tex]\( (6, 0) \)[/tex] (x-intercept) and [tex]\( (0, 8) \)[/tex] (y-intercept).
2. Draw the line [tex]\( 7x + 7y = 49 \)[/tex]. This line passes through [tex]\( (7, 0) \)[/tex] (x-intercept) and [tex]\( (0, 7) \)[/tex] (y-intercept).
3. Shade the region that satisfies both inequalities.
### Step 3: Identify the Corner Points
Determine the intersection points of the boundary lines within the feasible region. The feasible region for linear programming problems is typically a polygon, and the maximum or minimum values occur at the vertices (corner points) of this region.
To find the intersection of [tex]\( 8x + 6y = 48 \)[/tex] and [tex]\( 7x + 7y = 49 \)[/tex], solve the system of equations:
- [tex]\( 8x + 6y = 48 \)[/tex]
- [tex]\( 7x + 7y = 49 \)[/tex]
Multiply the second equation by [tex]\(\frac{6}{7}\)[/tex] to match the coefficients for [tex]\( y \)[/tex]:
[tex]\[ 6x + 6y = 42 \quad (scaled second equation) \][/tex]
Subtract this from the first equation:
[tex]\[ (8x + 6y) - (6x + 6y) = 48 - 42 \][/tex]
[tex]\[ 2x = 6 \implies x = 3 \][/tex]
Substitute [tex]\( x = 3 \)[/tex] back into the scaled second equation:
[tex]\[ 6(3) + 6y = 42 \implies 18 + 6y = 42 \implies 6y = 24 \implies y = 4 \][/tex]
Hence, the intersection point is [tex]\( (3, 4) \)[/tex].
### Step 4: Evaluate the Objective Function at Each Vertex
Evaluate [tex]\( P = 9x + 8y \)[/tex] at each vertex of the feasible region:
1. At [tex]\( (0, 0) \)[/tex]:
[tex]\[ P = 9(0) + 8(0) = 0 \][/tex]
2. At [tex]\( (6, 0) \)[/tex]:
[tex]\[ P = 9(6) + 8(0) = 54 \][/tex]
3. At [tex]\( (7, 0) \)[/tex]:
[tex]\[ P = 9(7) + 8(0) = 63 \][/tex]
4. At the intersection point [tex]\( (3, 4) \)[/tex]:
[tex]\[ P = 9(3) + 8(4) = 27 + 32 = 59 \][/tex]
### Step 5: Identify the Maximum Value
Based on these evaluations, the maximum value of [tex]\( P = 9x + 8y \)[/tex] within the feasible region is [tex]\( 59 \)[/tex], which occurs at the point [tex]\( (3, 4) \)[/tex].
Thus, the maximum value of [tex]\( P \)[/tex] is [tex]\(\boxed{59}\)[/tex], with the optimal point being [tex]\( x = 3 \)[/tex] and [tex]\( y = 4 \)[/tex].
