Let X = the number of the magazine covers,

    Y = the number of logos. 


Then the profit function is

P(X,Y) = 800*X + 500*Y      (1)    dollars.


The restrictions are 

4*X + 2*Y <= 24       (2)    (hours per week)

2*X + 0.5*Y <= 10     (3)    (hours per week)

Y <= 4X               (4)    ("no more than 4 logos for each magazine cover")


Other restrictions are non-negativity

X >= 0;  Y >= 0.


The feasible domain is shown in the plot below.



    


    Plot  4*X + 2*Y = 24 (red);  2*X + 0.5*Y = 10  (green);  and  Y = 4X (blue)



It is the quadrilateral in QI, adjacent to x-axis and bounded by the blue, red and green lines.


It has vertices 


    P1 = (2,8)       

    P2 = (4,4)    and

    P3 = (5,0)       (the green line x-intercept).


According to the Linear Programming method, we should calculate and compare the values of the profit function at these three points


    at P1:  P(2,8) = 800*2 + 500*8 = 5560 dollars;

    at P2:  P(4,4) = 800*4 + 500*4 = 5200 dollars.

    at P3:  P(5,0) = 800*5 + 500*0 = 4000 dollars.


The maximum value of the profit function is at P1.
It is the optimal solution.


Answer.  Optimal solution to the problem is 4 magazine covers and 8 logos.

         It satisfies the restrictions and gives maximal profit of 5560 dollars.