Linear Programming: Maximizing Objective Function
This problem is asking for the maximum value of the objective function \( 2x + y \) within the constraints described by the inequalities and the shaded feasible region on the graph. The constraints are:
1. \( y \geq 2x \) (Above the line \( y = 2x \))
2. \( y \geq x \) (Above the line \( y = x \))
3. \( x + y \leq 6 \) (Below the line \( x + y = 6 \))
The shaded region on the graph represents the set of all points (x, y) that satisfy these constraints. To find the maximum value of the objective function, we need to evaluate it at each of the vertices of the feasible region, as the maximum value in a linear programming problem always occurs at a vertex of the feasible region.
From the graph, it is clear that there are three vertices of the shaded feasible region:
1. \( P \) at the intersection of \( y = x \) and \( x + y = 6 \).
2. \( Q \) at (3, 3), which is provided in the problem statement.
3. \( R \) at the intersection of \( y = 2x \) and \( x + y = 6 \).
First, we need to find the coordinates of the vertex \( P \) and \( R \):
- For \( P \):
When \( y = x \), we substitute \( y \) for \( x \) in \( x + y = 6 \) to get \( x + x = 6 \) or \( 2x = 6 \), so that \( x = 3 \).
Thus, \( P \) is at (3, 3). The value of \( 2x + y \) at \( P \) is \( 2(3) + 3 = 9 \).
- For \( R \):
To find \( R \), solve the system of equations \( y = 2x \) and \( x + y = 6 \) simultaneously.
Replacing \( y \) in the second equation with \( 2x \) gives \( x + 2x = 6 \), which simplifies to \( 3x = 6 \), so \( x = 2 \).
Substituting back into \( y = 2x \), we find \( y = 4 \); thus, \( R \) is at (2, 4).
The value of \( 2x + y \) at \( R \) is \( 2(2) + 4 = 8 \).
We were given that \( Q \) is at (3, 3), which is the same point as \( P \), and we have already calculated the objective function value at \( P \), which is 9.
Now, compare the values of the objective function at these vertices:
- At \( P \) and \( Q \) (which are the same point), the value of \( 2x + y \) is 9.
- At \( R \), the value of \( 2x + y \) is 8.
The maximum value is therefore 9.
The correct answer to the problem is:
A: 9
