Although this problem is not clear, I did find what appears to be an identical problem shown below:
For a certain company, the cost function for producing x items is C(x)=40x+100 and the revenue function for selling x items is R(x)=−0.5(x−80)^2 +3,200. The maximum capacity of the company is 130 items. The profit function P(x) is the revenue function R(x) (how much it takes in) minus the cost function C(x) (how much it spends). In economic models, one typically assumes that a company wants to maximize its profit, or at least make a profit!
1. Assuming that the company sells all that it produces, what is the profit function?
P(x) = R(x) – C(x)
P(x) = (−0.5(x−80)^2 +3,200) – (40x+100)
P(x) = -0.5(x−80)^2 – 40x + 3,100
P(130) = -0.5(130-80)^2 – 40(130) + 3,100
P(130) = -1,250 – 5,200 + 3,100
P(130) = -3,350
2. What is the domain of P(x)? (Hint: Does calculating P(x) make sense when x=−10 or x=1,000?)
The domain of P(x) starts at 0 (no production) and goes to 130 (max production).
3. The company can choose to produce either 40 or 50 items. What is their profit for each case, and which level of production should they choose?
Profit when producing 40 items =
P(40) = -0.5(40-80)^2 – 40(40) + 3,100
P(40) = -800 – 1,600 + 3,100
P(40) = 700
Profit when producing 50 items =
P(50) = -0.5(50-80)^2 – 40(50) + 3,100
P(50) = -450 – 2,000 + 3,100
P(50) = 650
4. Can you please explain, our model why the company makes less profit when producing 10 more units?
The profit is a function of unit cost which is affected by supply and demand. If more units must be sold, their price must be discounted to sell them all. In our case, the drop in unit price does not cover the additional cost – therefore the total profit is less.
