Solution :
Given
P = 75 x + 40 y
The constraints are
[tex]$x+y \leq 18$[/tex] .........(i) and
[tex]$4x+y \leq 24$[/tex] .......(ii)
Solving (i) and (ii), we get
x = 2 and y = 16
Now
[tex]$x+y \leq 18$[/tex] (total arrangement less than equal to 18)
[tex]$4x+y \leq 24$[/tex] (total person hours less than equal to 24)
Maximum profit = 75 (2) + 40(16)
= 790
From the graph, the point of intersection and points that is plotted in between numbers between the 10 units on x-axis and y-axis are used in the graph.
So,
[tex]$x+y \leq 18$[/tex] became [tex]$x+y \geq 18$[/tex]
[tex]$4x+y \leq 24$[/tex] became [tex]$4x+y \geq 24$[/tex]
[tex]$x \geq 0$[/tex] became [tex]$x \leq 0$[/tex]
[tex]$y \geq 0$[/tex] became [tex]$y \leq 0$[/tex]
So now the profit at point (0, 18) = 75(0)+40(18) = 720
Profit at point (6,0) = 75(6)+40(0) = 450
Profit at intersection point (2,16) = 75(2)+40(16)
= 150 + 640
= 790
Maximum profit = 790
So maximum profit is attained when 2 wreath are built and sold and 18 center pieces are built and sold.
