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 + 0.003x2 and if the production capacity of the firm is at most 30,000 units in a specified time, how many units of penicillin must be manufactured and sold in that time to maximize the profit ?

Question

anonymous · Answer

Net profit = profit - expenses. For x units, net profit is 200*x - C(x). 

-500000-80x+200x=.003x^2
=> -500000+120x+-.003x^2
Take the derivative and set equal to zero to find a local max/min:

120-2*.003*x = 0 => x = 120/.002 = 20000.

Net profit when 20000 units are sold is -500000-80(20000)+200(20000)-.003(20000)^2 = 700000.

Plug in andpoints to determine if 700000 is a max or a min:

@0:-500000-80(0)+200(0)-.003(0)^2 = -500000

@30000: -500000-80(30000)+200(30000)-.003(30000)^2 = 400000

Both values are lower than 700000, so 700000 is a max and the solution (assuming I didn't make a stupid mistake somewhere)

anonymous · Answer

*CORRECTION:

-500000-80x+200x-.003x^2
=> -500000+120x-.003x^2

anonymous · Answer

Correction 2: Answer is 20,000, the number of production units at which net profit is 70,000.