Maximize the objective function M=6x+3y
under the constraints
{x ¡Ý 0
y ¡Ü 5
y ¡Ü -x + 10
y ¡Ý.5x - 4}
uh oh..the symbols didn't show..the iy is suppose to be the "greater than or equal to" symbol and the iU is suppose to be "less than or equal to symbol"
also there is multiple choice for the answer..
its either
a) no maximum exists
b) (10,0)
c) (8,0)
d) (28/3, 2/3)
To maximize the objective function M = 6x + 3y, we need to find the values of x and y that satisfy the given constraints. Let's solve this using a graphical method:
1. Plot the constraints on a graph:
- Draw a vertical line at x = 0.
- Draw a horizontal line at y = 5.
- Draw the line y = -x + 10.
- Draw the line y = 0.5x - 4.
2. Identify the feasible region:
- The shaded region where all the lines intersect represents the feasible region that satisfies all the constraints.
3. Determine the vertices of the feasible region:
- The vertices of the feasible region are the points where the lines intersect.
4. Evaluate the objective function at each vertex:
- Substitute the x and y values of each vertex into the objective function M = 6x + 3y.
- Compare the values to find the maximum.
5. Find the maximum value:
- The vertex that yields the highest value of M is the maximum value of the objective function.
By following these steps, you can find the maximum value of the objective function M = 6x + 3y under the given constraints.