Question

Based upon market research, the Hawthorne Company has determined that consumers are willing to purchase 135 units of their portable media player each week when the price is set at $26.10 per unit. At a unit price of $9.10, consumers are willing to buy 305 units per week.

Required:
a. Determine the weekly demand equation for this product, assuming price, p, and quantity, x, are linearly related.
b. Determine the weekly revenue function.
c. Determine the number of units consumers will demand weekly when the price is $93.00 per portable media player.
d. Determine the number of units consumers will demand weekly when the revenue is maximized.
e. Determine the price of each unit when the revenue is maximized

Answer 1

Answer:

a. P= -0.1x + 39.6

b.  R(x) = -0.1x^2 + 39.6x

c. x = -534 units

d. Number of units demand weekly when the revenue is maximized is 198 units

e. Price p = 15.8 units

Step-by-step explanation:

So for the demand equation let price =p

x= number of units sold

m = per unit price

b = initial fix amount

a. p = mx + b

When p = 26.10 $, x = 135 units so equation

26.10 = m(135) + b .......................(1)

When p = 9.10, x = 305 units so equation

9.10 = m(305) + b .......................(2)

subtracting equation (2) from equation (1)

26.10 - 9.10 =135x +b - 305x - b

17.00 = -170m

m= 17/-170

m= -0.1

Lets plug the value of m in the first equation

26.10 = m(135) + b

26.10 = (-0.1)(135) + b

26.10 = -13.5 + b

b= 26.10 + 13.5

b= 39.6

So the equation would be P= -0.1x + 39.6

b. Revenue = price * quantity

R(x) = p * x

R(x) = x (-0.1x + 39.6)

R(x) = -0.1x^2 + 39.6x

c. Here we have p = $ 93.00

P= -0.1x + 39.6

93 = -0.1x + 39.6

93 - 39.6 = -0.1x

-0.1x = 53.4

x = 53.4 / -0.1

x = -534 units

d. R(x) = -0.1x^2 + 39.6x

On differentiating it with respect to x.

R'(x) = -0.1(2)x^2-1 + 39.6x^1-1

R'(x) = -0.2x + 39.6

So for the maximum revenue differentiation of revenue function must be 0.

0 = -0.2x + 39.6

0.2x = 39.6

x = 39.6 / 0.2

x = 198 units

Number of units demand weekly when the revenue is maximized is 198 units

e. Price p = -0.1x + 39.6

on plugging the value x =238

Price p = -0.1(238) + 39.6

Price p = -23.8 + 39.6

Price p = 15.8 units