Question

The Fruity Juice Company has decided to produce and sell their juice Hairy Mango T M. They are about to begin production, but need to make one more big decision: How much to charge for each container. They want to maximize their revenue from the sale of the juice.

The research department has provided you with the following details about the demand of a similar juice drink sold in your desired market.

# Sold weekly Price (in cents)
5400 65
6000 50

(a) Find a formula for the price/demand function based on the data.
(b) Find a formula for the revenue function.
(c) Find the price they should charge to generate the maximum revenue, then calculate the maximum revenue. (Round to the nearest cent)

Answer 1

a) Price demand function = D = -40P + 8000

b) Revenue function = R = -40P² + 8000P

c) Price P = 100 cents and Revenue R = 4×10⁵ cents

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.