Answer:
To maximize the profit, the number of mountain bikes to be produced each week is 250 and the number of road bike to be produced each week is 0.
Step-by-step explanation:
Let, the number of mountain bike = x and number of road bike = y.
As minimum number of bike to be produced each week is 250.
Thus, x + y ≥ 250.
Also, mountain bike takes 4 hours to build and road bike takes 3 hours to build each week.
As, the employees work for 40 hours each week.
Thus, we get, 4x + 3y = 40.
Now, the objective is to maximize the profit given by z = 75x + 60y.
Thus, we get the system,
z = 75x + 60y
4x + 3y = 40
x + y ≥ 250
Using 'zero test' i.e. substituting ( 0,0 ) in the equation 2 and 3, we get that,
4x + 3y = 40 ⇒ 0 = 40, which is not true
x + y ≥ 250 ⇒ 0 ≥ 250, which is not true.
So, both the equations will have the solution region away from the origin as seen in the graph.
Also, the solution points are given by ( -710,960 ), ( 0,250 ) and ( 250,0 ).
Substituting these points in the objective function z = 75x + 60y, gives,
Points z = 75x + 60y
( -710,960 ) z = 75 × (-710) + 60 × 960 = 4,350
( 0,250 ) z = 75 × 0 + 60 × 250 = 15,000
( 250,0 ) z = 75 × 250 + 60 × 0 = 18,750
So, we see that the maximum value of the profit defined by z = 75x + 60y is $18,750 at the point ( 250,0 ).
Hence to maximize the profit, the number of mountain bikes to be produced each week is 250 and the number of road bike to be produced each week is 0.
