Answer :
To find the maximum value of the objective function [tex]\( P = 9x + 8y \)[/tex] subject to the given constraints, we need to first identify the feasible region defined by the constraints and then find the coordinates of the corner points of this region.
### Step 1: Write Down the Constraints
The constraints are:
1. [tex]\( 8x + 6y \leq 48 \)[/tex]
2. [tex]\( 7x + 7y \leq 49 \)[/tex]
3. [tex]\( x \geq 0 \)[/tex]
4. [tex]\( y \geq 0 \)[/tex]
### Step 2: Convert Inequalities to Equations
1. [tex]\( 8x + 6y = 48 \)[/tex]
2. [tex]\( 7x + 7y = 49 \)[/tex]
### Step 3: Find the Intersection Points of the Lines
To find the corner points, we solve these equations in pairs.
#### Intersection of [tex]\( 8x + 6y = 48 \)[/tex] and [tex]\( 7x + 7y = 49 \)[/tex]
1. Simplify the second equation:
[tex]\[ 7x + 7y = 49 \][/tex]
[tex]\[ x + y = 7 \][/tex]
2. Solve the system of linear equations:
[tex]\[ 8x + 6y = 48 \][/tex]
[tex]\[ x + y = 7 \][/tex]
From [tex]\( x + y = 7 \)[/tex], we get:
[tex]\[ y = 7 - x \][/tex]
Substitute [tex]\( y = 7 - x \)[/tex] in [tex]\( 8x + 6y = 48 \)[/tex]:
[tex]\[ 8x + 6(7 - x) = 48 \][/tex]
[tex]\[ 8x + 42 - 6x = 48 \][/tex]
[tex]\[ 2x + 42 = 48 \][/tex]
[tex]\[ 2x = 6 \][/tex]
[tex]\[ x = 3 \][/tex]
Now, substitute [tex]\( x = 3 \)[/tex] in [tex]\( x + y = 7 \)[/tex]:
[tex]\[ 3 + y = 7 \][/tex]
[tex]\[ y = 4 \][/tex]
Thus, the intersection point is [tex]\( (3, 4) \)[/tex].
### Step 4: Find Other Corner Points
#### Intersections with the Axes
1. Intersection with [tex]\( y \)[/tex]-axis ([tex]\( x = 0 \)[/tex]):
[tex]\[ 8(0) + 6y = 48 \][/tex]
[tex]\[ 6y = 48 \][/tex]
[tex]\[ y = 8 \][/tex]
Thus, the point is [tex]\( (0, 8) \)[/tex].
2. Intersection with [tex]\( x \)[/tex]-axis ([tex]\( y = 0 \)[/tex]):
[tex]\[ 8x + 6(0) = 48 \][/tex]
[tex]\[ 8x = 48 \][/tex]
[tex]\[ x = 6 \][/tex]
Thus, the point is [tex]\( (6, 0) \)[/tex].
#### Intersection with [tex]\( y \)[/tex]-axis ([tex]\( x = 0 \)[/tex]) for the second constraint:
[tex]\[ 7x + 7y = 49 \][/tex]
[tex]\[ 7(0) + 7y = 49 \][/tex]
[tex]\[ 7y = 49 \][/tex]
[tex]\[ y = 7 \][/tex]
Thus, the point is [tex]\( (0, 7) \)[/tex].
### Step 5: Check If Points are within the Feasible Region
The corner points are:
1. [tex]\( (0, 0) \)[/tex]
2. [tex]\( (6, 0) \)[/tex]
3. [tex]\( (3, 4) \)[/tex]
4. [tex]\( (0, 8) \)[/tex]
5. [tex]\( (0, 7) \)[/tex]
After checking which points satisfy all the constraints, we find that the relevant points for evaluation are [tex]\((6, 0)\)[/tex], [tex]\((3, 4)\)[/tex], and [tex]\((0, 7)\)[/tex].
### Step 6: Evaluate the Objective Function at the Corner Points
Now we evaluate [tex]\( P = 9x + 8y \)[/tex] at each corner point:
1. At [tex]\( (6, 0) \)[/tex]:
[tex]\[ P = 9(6) + 8(0) = 54 \][/tex]
2. At [tex]\( (3, 4) \)[/tex]:
[tex]\[ P = 9(3) + 8(4) = 27 + 32 = 59 \][/tex]
3. At [tex]\( (0, 7) \)[/tex]:
[tex]\[ P = 9(0) + 8(7) = 0 + 56 = 56 \][/tex]
### Step 7: Determine the Maximum Value
The maximum value of [tex]\( P = 9x + 8y \)[/tex] is achieved at the point [tex]\((3, 4)\)[/tex] and is equal to [tex]\( 59 \)[/tex].
Therefore, the corner points are [tex]\((3, 4)\)[/tex] and the maximum value of [tex]\( P \)[/tex] is 59.
### Step 1: Write Down the Constraints
The constraints are:
1. [tex]\( 8x + 6y \leq 48 \)[/tex]
2. [tex]\( 7x + 7y \leq 49 \)[/tex]
3. [tex]\( x \geq 0 \)[/tex]
4. [tex]\( y \geq 0 \)[/tex]
### Step 2: Convert Inequalities to Equations
1. [tex]\( 8x + 6y = 48 \)[/tex]
2. [tex]\( 7x + 7y = 49 \)[/tex]
### Step 3: Find the Intersection Points of the Lines
To find the corner points, we solve these equations in pairs.
#### Intersection of [tex]\( 8x + 6y = 48 \)[/tex] and [tex]\( 7x + 7y = 49 \)[/tex]
1. Simplify the second equation:
[tex]\[ 7x + 7y = 49 \][/tex]
[tex]\[ x + y = 7 \][/tex]
2. Solve the system of linear equations:
[tex]\[ 8x + 6y = 48 \][/tex]
[tex]\[ x + y = 7 \][/tex]
From [tex]\( x + y = 7 \)[/tex], we get:
[tex]\[ y = 7 - x \][/tex]
Substitute [tex]\( y = 7 - x \)[/tex] in [tex]\( 8x + 6y = 48 \)[/tex]:
[tex]\[ 8x + 6(7 - x) = 48 \][/tex]
[tex]\[ 8x + 42 - 6x = 48 \][/tex]
[tex]\[ 2x + 42 = 48 \][/tex]
[tex]\[ 2x = 6 \][/tex]
[tex]\[ x = 3 \][/tex]
Now, substitute [tex]\( x = 3 \)[/tex] in [tex]\( x + y = 7 \)[/tex]:
[tex]\[ 3 + y = 7 \][/tex]
[tex]\[ y = 4 \][/tex]
Thus, the intersection point is [tex]\( (3, 4) \)[/tex].
### Step 4: Find Other Corner Points
#### Intersections with the Axes
1. Intersection with [tex]\( y \)[/tex]-axis ([tex]\( x = 0 \)[/tex]):
[tex]\[ 8(0) + 6y = 48 \][/tex]
[tex]\[ 6y = 48 \][/tex]
[tex]\[ y = 8 \][/tex]
Thus, the point is [tex]\( (0, 8) \)[/tex].
2. Intersection with [tex]\( x \)[/tex]-axis ([tex]\( y = 0 \)[/tex]):
[tex]\[ 8x + 6(0) = 48 \][/tex]
[tex]\[ 8x = 48 \][/tex]
[tex]\[ x = 6 \][/tex]
Thus, the point is [tex]\( (6, 0) \)[/tex].
#### Intersection with [tex]\( y \)[/tex]-axis ([tex]\( x = 0 \)[/tex]) for the second constraint:
[tex]\[ 7x + 7y = 49 \][/tex]
[tex]\[ 7(0) + 7y = 49 \][/tex]
[tex]\[ 7y = 49 \][/tex]
[tex]\[ y = 7 \][/tex]
Thus, the point is [tex]\( (0, 7) \)[/tex].
### Step 5: Check If Points are within the Feasible Region
The corner points are:
1. [tex]\( (0, 0) \)[/tex]
2. [tex]\( (6, 0) \)[/tex]
3. [tex]\( (3, 4) \)[/tex]
4. [tex]\( (0, 8) \)[/tex]
5. [tex]\( (0, 7) \)[/tex]
After checking which points satisfy all the constraints, we find that the relevant points for evaluation are [tex]\((6, 0)\)[/tex], [tex]\((3, 4)\)[/tex], and [tex]\((0, 7)\)[/tex].
### Step 6: Evaluate the Objective Function at the Corner Points
Now we evaluate [tex]\( P = 9x + 8y \)[/tex] at each corner point:
1. At [tex]\( (6, 0) \)[/tex]:
[tex]\[ P = 9(6) + 8(0) = 54 \][/tex]
2. At [tex]\( (3, 4) \)[/tex]:
[tex]\[ P = 9(3) + 8(4) = 27 + 32 = 59 \][/tex]
3. At [tex]\( (0, 7) \)[/tex]:
[tex]\[ P = 9(0) + 8(7) = 0 + 56 = 56 \][/tex]
### Step 7: Determine the Maximum Value
The maximum value of [tex]\( P = 9x + 8y \)[/tex] is achieved at the point [tex]\((3, 4)\)[/tex] and is equal to [tex]\( 59 \)[/tex].
Therefore, the corner points are [tex]\((3, 4)\)[/tex] and the maximum value of [tex]\( P \)[/tex] is 59.
