The problem is a Linear Programming problem where we are tasked with finding the maximum value of an objective function, subject to a set of constraints. Here we are trying to find values of 'x' and 'y' which would maximize the function p = 3x + 2y, given the constraints:
1.6*x + 0.8*y ≤ 8
0.13*x + 0.26*y ≤ 1.3
8*x + 8*y ≤ 48
x ≥ 0
y ≥ 0
Here are the detailed step-by-step solutions:
Step 1: Start by plotting the constraint 1.6*x + 0.8*y ≤ 8;
For x = 0, y = 10, and for y = 0, x = 5
Plot these points and shade all the possible solutions which satisfy this inequality.
Step 2: Repeat the process for other constraints:
For 0.13*x + 0.26*y ≤ 1.3, for x=0, y=5, and for y=0, x=10. Plot these points and the region of solutions.
For 8*x + 8*y ≤ 48, for x=0, y=6, and for y=0, x=6. Plot these points and the region of solutions.
For x ≥ 0, plot a vertical line through the point (0,0) and shade the right side of it.
For y ≥ 0, plot a horizontal line through the point (0,0) and shade the upper side of it.
Step 3: The feasible region is determined by the intersection of the regions that satisfy all these constraints.
Step 4: Take the objective function p = 3x + 2y. Plot different objective function lines by substituting different values for 'p', the most common choice is to select the value where both x and y are zero.
Step 5: Move the line of the objective function outwards from the origin (0,0) until the last point of intersection with the feasible region. This point will give the maximum value of the objective function given the constraints, this is because linear programming operates under the assumption that the optimum value must occur at an extreme point (vertex) of the feasible region.
Step 6: The coordinates of this last point of intersection are the 'x' and 'y' values that maximize the function p = 3x + 2y under the given constraints. Substitute these values in the equation p = 3x + 2y to find the maximum value of 'p'.
This is a feasible method to solve the problem in lieu of programming solution. You would need graph paper or a software capable of doing the same to accurately map out the feasible region and determine the point of optimality.