Maximize P=3x+2y

subject to 6x+3y<=24
3x+6y=<30
X,Y>=10

That x,y>=10 is a killer.

Removing that, we get a max at
p(2,4) = 14

A good online tool is at

http://www.zweigmedia.com/RealWorld/LPGrapher/lpg.html

To maximize the objective function P=3x+2y, subject to the given constraints, we can follow these steps:

1. Graph the feasible region:
- Start by graphing the two inequalities 6x+3y<=24 and 3x+6y<=30.
- Convert the inequalities to equations: 6x+3y=24 and 3x+6y=30.
- Plot the lines on a coordinate plane.
- Shade the region that satisfies both inequalities.
- Note that the constraints X>=10 and Y>=10 place additional restrictions on the feasible region.

2. Identify the vertices of the feasible region:
- Find the points where the lines intersect within the feasible region.
- These points represent the vertices of the feasible region.

3. Evaluate the objective function at each vertex:
- Substitute the coordinates of each vertex into the objective function P=3x+2y.
- Calculate the value of P at each vertex.

4. Determine the maximum value of P:
- Compare the values of P at each vertex.
- The vertex that yields the highest value of P is the solution to the problem.
- This vertex represents the optimal value for x and y to maximize P.

By following these steps, you will be able to find the maximum value of P=3x+2y while satisfying the given constraints.