Question 1119561
.
<pre>
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.



    {{{drawing(660, 660, -1.5, 10.5, -1.5, 25.5,
            grid(1),
graph( 660, 660, -1.5, 10.5, -1.5, 25.5,
          (24-4x)/2,  2*(10-2x), 4x)
)}}}


    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.


<U>Answer</U>.  Optimal solution to the problem is 4 magazine covers and 8 logos.

         It satisfies the restrictions and gives maximal profit of 5560 dollars.
</pre>

Solved.


============


To see other similar problems solved by the Linear Programming method, look into the lesson

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/misc/Solving-minimax-problems-by--the-Linear-Programming-method.lesson>Solving minimax problems by the Linear Programming method</A> 

in this site.