Let X be the number of downhill skis, and

let Y be the number of cross-country skis.


Then the manufacturing time to produce X downhill skis and Y cross-country skis is  3X + 7Y  hours.

and  the finishing     time to produce X downhill skis and Y cross-country skis is  2X + 2Y  hours.


Therefore, the two major constraints are


    3X + 7Y <= 35 hours    (manufacturing time)     (1)

    2X + 2Y <= 18 hours    (finishing     time)     (2)


Add to it the standard non-negativity constraints

    X >= 0,  Y >= 0.                                (3)


Now we have everything to construct the feasibility region.


For it, you draw the lines  3X + 7Y = 35  and  2X + 2Y = 18  from the constraints.


They are shown in the figure below.


    


                Plot y =   (red)  and  y =  (green)



Inequalities (3) define the first quadrant QI.



The solution set to the given inequalities (the feasibility domain) are the points of the coordinate plane, 

    that are in QI and belong the quadrilateral below (or on) the red line and below (or on) the green line.



Again: the feasibility domain is the quadrilateral in QI restricted by the red and by the green lines, 
adjacent to coordinate axes, including its sides and vertices.


The corner points are


    1)  P1 = (0,0)   the origin of the coordinate plane

    2)  P2 = (0,5)   Y-intercept of the red line       

    3)  P3 = (7,2)   intersection of the red line and the green line 

    4)  P4 = (9,0)   X-intercept of the green line  


At this point, I completed my explanations regarding the corner points.


Next, to solve this minimax problem,  compare the values of the objective function z = 62X + 63Y at the corner points.


    P1:  z = 62*0 + 63*0 = 0

    P2:  z = 62*0 + 63*5 = 315

    P3:  z = 62*7 + 63*2 = 560

    P4:  z = 62*9 + 63*0 = 558


You see that the profit is maximum at point P3.


It gives the solution to the problem : X= 7 downhill skis,  Y= 2 cross-country skis with the maximum profit of 560 dollars.    ANSWER