Answer:
75 $
Step-by-step explanation:
According to problem statement p(300) = 600
And we know that with a rebate of 40 $, numbers of units sold will increase by 80 then if x is number of units sold, the increase in units is
( x - 300 ) , and the price decrease
(1/80)*40 = 0,5
Then the demand function is:
D(x) = 600 - 0,5* ( x - 300 ) (1)
And revenue function is:
R(x) = x * (D(x) ⇒ R(x) = x* [ 600 - 0,5* ( x - 300 )]
R(x) = 600*x - 0,5*x * ( x - 300 )
R(x) = 600*x - 0,5*x² - 150*x
R(x) = 450*x - (1/2)*x²
Now taking derivatives on both sides of the equation we get
R´(x) = 450 - x
R´(x) = 0 ⇒ 450 - x = 0
x = 450 units
We can observe that for 0 < x < 450 R(x) > 0 then R(x) has a maximum for x = 450
Plugging this value in demand equation, we get the rebate for maximize revenue
D(450) = 600 - 0,5* ( x - 300 )
D(450) = 600 - 225 + 150
D(450) =
D(450) = 600 - 0,5*( 150)
D(450) = 600 - 75
D(450) = 525
And the rebate must be
600 - 525 = 75 $
