I am having trouble on a math problem. I have tried everything I know to do, but I cannot seem to find the answer. (Well, I get an answer, but I do not think it is correct).
I will write out the problem, show you my work, and my answer. Tell me your thoughts.
The cost per unit for the production of a certain radio model is $60. The manufacturer charges $90 per unit for orders of 100 or less. To encourage large orders, the manufacturer reduces the charge by $0.15 per radio for each unit ordered in excess of 100.
a. Write the profit as a function of x.
b. How many radios should he sell to maximize profit?
So there are a few different functions here.
The cost to produce a radio is $60. Therefore:
c ( x ) = 60 x
The revenue the guy gets from selling radios is dependent on how many radios he sells. If it is less than 100 radios, it is $90 per radio, otherwise it is $90 per radio minus .15 cents per every radio over 100.
f ( x ) =
if x <= 100
then return 90x
else if x > 100
then return 90x - .15(x - 100)
Profit is revenue minus cost. Therefore:
p(x) = f(x) - c(x)
Therefore, if we are under 100 radios, p(x) is this:
90x - 60x
If we are above 100 radios:
(90x - .15(x-100)) - 60x
(90x - .15x + 15) - 60x
(89.85x + 15) - 60x
p(x) = 29.85x + 15
This function is completely linear. It has no max and no min. Profit never maximizes. It approaches infinity.
Am I doing something wrong?