This is a linear programming problem in which we have given resources (oven and decorating room) and production options (cakes, x and cookies y).
The objective is to maximize profit, given by Z=25x+40y.
To analyze the problem, you would need to identify the constraints, namely
2x+3y/2 ≤ 15, or
y ≤ (2/3)(15-2x), and
3x+2y/3 ≤ 13, or
y ≤ (3/2)(13-3x)
Plot these as lines (equality) and consider only integers (batches).
The two lines will intersect at the point (4,6) which represents the optimal use of the resources, but does not necessarily mean the maximum profit.
The profit function is given by:
and has to be maximized within the feasible region.
Points outside the figure are non-feasible because the constraints are violated.
Any point within the figure enclosed by the two lines and the axes is a feasible solution, as long as the points are in the integer domain.
To find the maximum or minimum profit, we do not need to check the interior points, but we need to check all points at or close to corners of the polygon.
The one that gives the maximum value of z(x,y) is the combination of (integer) batches of each kind.
Here's a link to the graph:
If you need further help, please post.