Answer :
To find the minimum value of [tex]\( C = 7x + 8y \)[/tex] given the constraints, we will follow a series of steps, including identifying the feasible region and evaluating the objective function at the vertices of the feasible region.
### Step 1: Identify the constraints
The given constraints are:
1. [tex]\(2x + y \geq 0\)[/tex]
2. [tex]\(x + y \geq 0\)[/tex]
3. [tex]\(x \geq 0\)[/tex]
4. [tex]\(y \geq 0\)[/tex]
### Step 2: Determine the feasible region
To find the feasible region, we need to plot the inequalities and identify the region that satisfies all of them.
#### Inequality 1: [tex]\(2x + y \geq 0\)[/tex]
Rewriting this in slope-intercept form, we get:
[tex]\[ y \geq -2x \][/tex]
#### Inequality 2: [tex]\(x + y \geq 0\)[/tex]
Rewriting this in slope-intercept form, we get:
[tex]\[ y \geq -x \][/tex]
#### Inequality 3: [tex]\( x \geq 0 \)[/tex]
This inequality indicates that the feasible region lies to the right of the [tex]\( y \)[/tex]-axis.
#### Inequality 4: [tex]\( y \geq 0 \)[/tex]
This inequality indicates that the feasible region lies above the [tex]\( x \)[/tex]-axis.
### Step 3: Find the intersection points (vertices)
The intersection points of these inequalities within the first quadrant (since [tex]\( x \geq 0 \)[/tex] and [tex]\( y \geq 0 \)[/tex]) will define the feasible region. Calculate the intersection points:
1. Intersection of [tex]\( y = -2x \)[/tex] and [tex]\( y = -x \)[/tex]:
- Set [tex]\( -2x = -x \)[/tex] leading to [tex]\( x = 0 \)[/tex].
- Substitute [tex]\( x = 0 \)[/tex] in [tex]\( y = -x \)[/tex] to get [tex]\( y = 0 \)[/tex].
- Intersection point: [tex]\( (0,0) \)[/tex].
2. Intersection of [tex]\( y = -2x \)[/tex] and [tex]\( x = 0 \)[/tex]:
- Set [tex]\( x = 0 \)[/tex] directly leading to [tex]\( y = 0 \)[/tex].
- Intersection point: [tex]\( (0, 0) \)[/tex] (same point as before).
3. Intersection of [tex]\( y = -x \)[/tex] and [tex]\( y = 0 \)[/tex]:
- Set [tex]\( y = 0 \)[/tex] leading to [tex]\( x = 0 \)[/tex].
- Intersection point: [tex]\( (0,0) \)[/tex] (same point as before).
Since the constraints only permit [tex]\( x \geq 0 \)[/tex] and [tex]\( y \geq 0 \)[/tex], and both [tex]\( y = -2x \)[/tex] and [tex]\( y = -x \)[/tex] are negative slopes intersecting at the origin, we should explore within the feasible region in the first quadrant formed solely by points satisfying both inequalities without violating [tex]\( x \geq 0 \)[/tex] and [tex]\( y \geq 0\)[/tex].
### Step 4: Feasible points
We have deduced that [tex]\( (0, 0) \)[/tex] is the single feasible vertex here within the quadrant:
### Step 5: Evaluate the objective function at each vertex
- At [tex]\( (0,0) \)[/tex]:
[tex]\[ C = 7(0) + 8(0) = 0 \][/tex]
### Conclusion
After evaluating the objective function at all feasible points, we find that the minimum value of [tex]\( C = 7x + 8y \)[/tex] with the given constraints is:
[tex]\[ \boxed{0} \][/tex]
Given the provided multiple-choice options (32, 42, 46, 64), none of these align correctly as the step-by-step analysis yields [tex]\(0\)[/tex], which might imply either a different set of constraints or simpler misinterpretation. Double-checking initial steps and constraints verifies [tex]\( \boxed{0} \)[/tex].
### Step 1: Identify the constraints
The given constraints are:
1. [tex]\(2x + y \geq 0\)[/tex]
2. [tex]\(x + y \geq 0\)[/tex]
3. [tex]\(x \geq 0\)[/tex]
4. [tex]\(y \geq 0\)[/tex]
### Step 2: Determine the feasible region
To find the feasible region, we need to plot the inequalities and identify the region that satisfies all of them.
#### Inequality 1: [tex]\(2x + y \geq 0\)[/tex]
Rewriting this in slope-intercept form, we get:
[tex]\[ y \geq -2x \][/tex]
#### Inequality 2: [tex]\(x + y \geq 0\)[/tex]
Rewriting this in slope-intercept form, we get:
[tex]\[ y \geq -x \][/tex]
#### Inequality 3: [tex]\( x \geq 0 \)[/tex]
This inequality indicates that the feasible region lies to the right of the [tex]\( y \)[/tex]-axis.
#### Inequality 4: [tex]\( y \geq 0 \)[/tex]
This inequality indicates that the feasible region lies above the [tex]\( x \)[/tex]-axis.
### Step 3: Find the intersection points (vertices)
The intersection points of these inequalities within the first quadrant (since [tex]\( x \geq 0 \)[/tex] and [tex]\( y \geq 0 \)[/tex]) will define the feasible region. Calculate the intersection points:
1. Intersection of [tex]\( y = -2x \)[/tex] and [tex]\( y = -x \)[/tex]:
- Set [tex]\( -2x = -x \)[/tex] leading to [tex]\( x = 0 \)[/tex].
- Substitute [tex]\( x = 0 \)[/tex] in [tex]\( y = -x \)[/tex] to get [tex]\( y = 0 \)[/tex].
- Intersection point: [tex]\( (0,0) \)[/tex].
2. Intersection of [tex]\( y = -2x \)[/tex] and [tex]\( x = 0 \)[/tex]:
- Set [tex]\( x = 0 \)[/tex] directly leading to [tex]\( y = 0 \)[/tex].
- Intersection point: [tex]\( (0, 0) \)[/tex] (same point as before).
3. Intersection of [tex]\( y = -x \)[/tex] and [tex]\( y = 0 \)[/tex]:
- Set [tex]\( y = 0 \)[/tex] leading to [tex]\( x = 0 \)[/tex].
- Intersection point: [tex]\( (0,0) \)[/tex] (same point as before).
Since the constraints only permit [tex]\( x \geq 0 \)[/tex] and [tex]\( y \geq 0 \)[/tex], and both [tex]\( y = -2x \)[/tex] and [tex]\( y = -x \)[/tex] are negative slopes intersecting at the origin, we should explore within the feasible region in the first quadrant formed solely by points satisfying both inequalities without violating [tex]\( x \geq 0 \)[/tex] and [tex]\( y \geq 0\)[/tex].
### Step 4: Feasible points
We have deduced that [tex]\( (0, 0) \)[/tex] is the single feasible vertex here within the quadrant:
### Step 5: Evaluate the objective function at each vertex
- At [tex]\( (0,0) \)[/tex]:
[tex]\[ C = 7(0) + 8(0) = 0 \][/tex]
### Conclusion
After evaluating the objective function at all feasible points, we find that the minimum value of [tex]\( C = 7x + 8y \)[/tex] with the given constraints is:
[tex]\[ \boxed{0} \][/tex]
Given the provided multiple-choice options (32, 42, 46, 64), none of these align correctly as the step-by-step analysis yields [tex]\(0\)[/tex], which might imply either a different set of constraints or simpler misinterpretation. Double-checking initial steps and constraints verifies [tex]\( \boxed{0} \)[/tex].
