Answer:
a) Price demand function = D = -40P + 8000
b) Revenue function = R = -40P² + 8000P
c) Price P = 100 cents and Revenue R = 4×10⁵ cents
Step-by-step explanation:
We are given quantity and price of Hairy Mango T M juice
Let P represents price in cents and D represents demand
D₁ = 5400
D₂ = 6000
P₁ = 65
P₂ = 50
(a) Find a formula for the price/demand function based on the data.
The demand and price relationship can be modeled as a linear relationship,
D = aP + b
Substitute the given data points into above equation
5400 = 65a + b eq. 1
6000 = 50a + b eq. 2
Solving eq. 1 and eq, 2 simultaneously,
b = 6000 - 50a
5400 = 65a + (6000 - 50a)
5400 - 6000 = 65a - 50a
-600 = 15a
a = -600/15
a = -40
b = 6000 - 50a
b = 6000 - 50(-40)
b = 6000 + 2000
b = 8000
So the price demand function becomes,
D = aP + b
D = -40P + 8000
(b) Find a formula for the revenue function.
The revenue function is given by
R = D*P
R = (-40P + 8000)*P
R = -40P² + 8000P
(c) Find the price they should charge to generate the maximum revenue, then calculate the maximum revenue. (Round to the nearest cent)
Maximum revenue is obtained when
dR/dp = 0
dR/dp = -40P² + 8000P
dR/dp = -80P + 8000
0 = -80P + 8000
80P = 8000
P = 8000/80
P = 100 cents
The maximum revenue is
R = -40(100)² + 8000(100)
R = -400000 + 800000
R = 400000 cents
R = 4×10⁵ cents
Therefore, the maximum revenue is 4×10⁵ cents when the price is set to 100 cents.
