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 120 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! Assuming that the company sells all that it produces, what is the profit function? What is the domain of P(x)? Hint: Does calculating P(x) make sense when x=−10 or x=1,000? 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 = Profit when producing 50 items = Can you explain, from our model, why the company makes less profit when producing 10 more units?

Question

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 120 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!

Assuming that the company sells all that it produces, what is the profit function?

What is the domain of P(x)?
Hint: Does calculating P(x) make sense when x=−10 or x=1,000?
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 =

Profit when producing 50 items =