Question
If x thousand dollars is spent on labor and y thousand dollars is spent on euqpment,The output of a certain factory will be Q(x,y)=2x^2+y^2+6y units.If 120,000 dollars is available,how should this be allocated between labor and equipment
to generate the largest possible output
to generate the largest possible output
Answers
Steve
for max Q, we need
∂Q/∂x=0 and ∂Q∂y=0
∂Q/∂x = 4x
∂Q/∂y = 2y+6
so, there's a max/min at (0,-3).
Since the 2nd partials are both positive, it's a minimum.
So, we need to maximize
2x^2+y^2+6y subject to
x+y = 120
Q(0,120) = 15120
Q(120,0) = 28800
Q(119,1)
Clearly, since 2x^2 decreases faster than y^2+6y increases for each dollar moved from x to y, the max is Q(120,0).
Doesn't seem right, since some equipment must be used.
∂Q/∂x=0 and ∂Q∂y=0
∂Q/∂x = 4x
∂Q/∂y = 2y+6
so, there's a max/min at (0,-3).
Since the 2nd partials are both positive, it's a minimum.
So, we need to maximize
2x^2+y^2+6y subject to
x+y = 120
Q(0,120) = 15120
Q(120,0) = 28800
Q(119,1)
Clearly, since 2x^2 decreases faster than y^2+6y increases for each dollar moved from x to y, the max is Q(120,0).
Doesn't seem right, since some equipment must be used.