The quantity demanded each month of a cell phone cover manuf

The quantity demanded each month of a cell phone cover, manufactured by Cell-R-Us, is related to the price per cover. The equation p = 0.00042x + 6 where p denotes the unit price in dollars and x is the number of covers demanded, relates the demand to the price. The total monthly cost (in dollars) for producing the x number of covers is given by C(x) = 600 + 2x 0.00002x 2 . How many cell phone covers should the company produce each month in order to maximize its profit?

Solution

p(x) = -0.00042x + 6 ;p denotes the unit price in dollars and x is the number of covers demanded

total monthly cost , C(x) = 600 +2x - 0.00002x^2

Demand price for x covers : P(x) = x(-0.00042x +6) = 6x - 0.00042x^2

Profit = P(x) - C(x)

= 6x - 0.00042x^2 - (600 +2x - 0.00002x^2)

= 4x - 0.0004x^2 - 600

Maximize profit : maximum profit would occurs at vertex x = -b/2a = - ( 4/(2*(-0.0004) )

x = 4/0.0008 = 5000 covers in order to maximize its profit