6 the revenue from sales of x units of a product is given by

6) the revenue from sales of x units of a product is given by R(x)=200x - .01x(squared), and the cost of producing and selling the product can be described by C(x)=38x + .01x(squared) + 16,000.

a. Write the profit function P(x)

b. Producing and selling how many units will give maxium profit?

c. What is the maximum possible profit for the product?

please show all work!

Solution

a] Profit = Revenue - Total Cost

=> P(x) = 200x - 0.01x² - 38x - 0.01x² - 16,000

=> P(x) = 162x - 0.02x² - 16000.

b] To maximize profits, differentiate P(x) and equate it to zero.

P \' (x) = 162 - 0.04x

P \' (x) = 0 [for maximum P(x)]

=> 0 = 162 - 0.04x

=> x = 4050.

c] for x = 4050, P(x) = 162(4050) - 0.02(4050)² - 16000 = 312,050.