Respuesta :
Answer:
They should produce 15 snowboards and 28 skis in a week to maximize their profit.
Step-by-step explanation:
Let x be the number of snowboards and y be the number of skis.
It takes 8 hours to manufacture a snowboard and 10 hours to manufacture a set of skis, there are 400 hours of labor available each week for production.
[tex]8x+10y\leq 400[/tex]
Due to demand, they must produce at least 15 snowboards.
[tex]x\geq 15[/tex]
Snowboards profit $55 each and skis profit $80 each.
[tex]profit=55x+80y[/tex]
The required linear programming problem is
Maximize profit [tex]Z=55x+80y[/tex]
S.t.,
[tex]8x+10y\leq 400[/tex]
[tex]x\geq 15[/tex]
[tex]y\geq 0[/tex]
Sketch the graph of all inequalities as shown below.
In inequality [tex]8x+10y\leq 400[/tex], then sign of inequality is ≤. Check the equality by (0,0).
[tex]8(0)+10(0)\leq 400[/tex]
[tex]0\leq 400[/tex]
This inequality is true for (0,0). It means the shaded region of this inequality is below the line.
The shaded region of x≥15 is right side of vertical line x=15.
The shaded region of y≥0 is above of horizontal line y=0.
From the figure it is clear that the extreme point of common shaded region are (15,0), (15,28) and (50,0).
Find the value of profit function at extreme points.
At (15,0),
[tex]Z=55(15)+80(0)=825[/tex]
At (15,28),
[tex]Z=55(15)+80(28)=3065[/tex]
At (50,0),
[tex]Z=55(50)+80(0)=2750[/tex]
The profit is maximum at (15,28).
Therefore, they should produce 15 snowboards and 28 skis in a week to maximize their profit.
