For a certain company, the cost function for producing x items is C(x)=30x+200 and the revenue function for selling x items is R(x)=0.5(x100)^2+5,000. The maximum capacity of the company is 160 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)=

   Hint: Profit = Revenue - Cost as we examined in Discussion 3.

2. What is the domain of P(x)?

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

3. The company can choose to produce either 70 or 80 items. What is their profit for each case, and which level of production should they choose?

   Profit when producing 70 items =

   Profit when producing 80 items =

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