Help me determine the profit when a number of items is given

Question

Scenario: For a certain company, the cost function for producing x items is C(x)=30x+150 and the revenue function for selling x items is R(x)=−0.5(x−90)^2+4,050. The maximum capacity of the company is 150 items.

I already know the profit function:

P(x) = R(x)-C(x)

P(x)= -0.5(x-90)^2+4,050-(30x)+150

P(x)= -0.5x^2+60x-150

What is the domain of P(x)?

-wouldn't the domain of P(x) be 0?

Hint: Does calculating P(x) make sense when x=−10 or x=1,000?

The company can choose to produce either 60 or 70 items. What is their profit for each case, and which level of production should they choose?

I just don't know how to calculate the profit when a number of items is given.

Profit when producing 60 items = ?

Profit when producing 70 items = ?

Can you explain, from our model, why the company makes less profit when producing 10 more units?

If Someone could help me I would greatly appreciate it.

Thanking you in advance,

Michael Pagano

Answer 1

P(60) = .-5(3600) + 3600 -150 = 1800-150 = 1650

P(70) = -.5(70)^2 + 60(70) - 150 = -2450 +4200 - 150 = 1600

P'(x) =- x + 60 = 0

x=60 is the profit maximizing output level. at output more or less than 60, profit is less. If you produce more than 60, while revenue goes up, costs go up more than the revenue increase, causing profit to decrease.

the domain any integer 0 to 150 (the maximum capacity) , as x can be any non-negative integer

where 0<x< 150

x = 1000 would make sense, except it's not practical, given the maximum capacity of 150

even though you make less profit at 70 than 60, it might be a feasible strategy to gain a larger market share and drive your competitors out of business, such that long term profits might be larger, if you sacrifice some short term profits.