Question

A company sells a product A. If the product price is $30, then the company will be able to
sell 3000 items. For every $5 the company lowers the price, then it will be able to sell 1000
more. Suppose that the company has a fixed cost of $20000, and that the production cost
per item is $5. Find the price to set per item in order to maximize the profit. (Note: The
profit is given by the revenue minus the costs. The revenue is the price per item times the
number of sold items.)

Answer 1

Let x = # of units sold, and p = price per unit
first we can find an equation for price , in terms of units sold.
(3000 sold, $30) ( 4000 sold, $25)
we can make a line using slope intercept formula
p = -1/200 * x + 45

Revenue = x * p
= x ( -1/200 * x + 45)
= -1/200 * x^2 + 45x

Profit = Revenue - costs
Profit = (-1/200 * x^2 + 45x ) - ( 5x + 20000)

Answer 2

Profit = -1/200*x^2 + 40x - 20000
maximize x = -b/(2a)
= -40/ (2* -1/200 ) = 4000 . so selling 4000 units will maximize profits, and the price that corresponds to 4000 units sold is 25 dollars.