Question

A liquid form of penicillin manufactured by a pharmaceutical firm is sold in bulk at a price of
$200 per unit. If the total production cost (in dollars) for x units is
C(x) = 500,000 + 80x + 0.003x^2
and if the production capacity of the firm is at most 30,000 units in a specfied time, how many
units of penicillin must be manufactured and sold in that time to maximize the profit?

Answer 1

The revenue is given by the unit price * number of units.
Profit = Revenue - Cost = 200x - 500,000 - 80x - 0.003x^2 = 120x - 500,000 - 0.003x^2
For profit to be maximum dP/dx = 0
120 - 0.006x = 0
x = 120/0.006 = 20,000

Therefore, to maximize profit, the firm should produce 20,000 penicillin.