A company manufactures and sells x smartphones per week. The weekly price-demand and cost equations are p=500−0.5x and C(x)=20,000+135x respectively. a. What price should the company charge for the phones and how many phones should be produced to maximize the weekly revenue? What is the maximum weekly revenue ?
b. What is the maximum weekly profit ? How much should the company charge for the phones, and how many phones should be produced to realize the maximum weekly profit ?

a)

revenue = x amount of phones x price

revenue = x(500 - 0.5x)

revenue = 500x - 0.5x²

we find revenue' (derivative):

revenue' = 500 - x

x = 500

the company should sell 500 smartphones to maximize revenue, the selling price = 500 - (0.5 x 500) = $250 per smartphone. Maximum weekly revenue = $250 x 500 = $125,000

b)

profit = revenue - cost

profit = 500x - 0.5x²  - 20,000 - 135x

profit = -0.5x² + 365x - 20,000

we must find profit' (derivative):

profit' = -x + 365

x = 365

In order to maximize profits, you have to sell 365 smartphones per week. Maximum weekly profit = -0.5(365²) + 365(365) - 20,000 = -66,612.50 + 133,225 - 20,000 = $46,612.50.

The smartphone's price = 500 - (0.5 x 365) = $317.50

the definition of a small business is an independently owned and operated company that is limited in size and in revenue depending on the industry. a local bakery that employs 10 people is an example of a small business. a manufacturing facility that employees less than 500 people is an example of a small business.