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 specfied time, how many
units of penicillin must be manufactured and sold in that time to maximize the profit?

Answer 1

the function fot the profit will be the total selling price minus the total production cost.
f(x) = 200x - (500,000 + 80x +0.003x^2)
f(x) = -0.003x^2 + 120x -500,000

then to find the value of x which will result in maximum profit, you can use the derivative of f(x)
f'(x) = 0
f'(x) = -0.006x + 120 = 0
0.006x = 120
x = 20000

then plug x=20000 into f(x)
f(x) = -0.003(20000)^2 +120(20000) -500000
that's the maximum profit

Answer 2

oh the question is how many penicillin, the answer is 20000 units.
i thought you're asked to find the profit , haha. then just forget about plugging x=20000 into the equation