Math Profit Functions

Question

For a certain company, the cost function for producing x items is C(x)=50x+200 and the revenue function for selling x items is R(x)=−0.5(x−120)2+7,200. The maximum capacity of the company is 140 items.

1. Assuming that the company sells all that it produces, what is the profit function?
2. P(x)=

1. What is the domain of P(x)?

1. 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?
2. Profit when producing 70 items =   
3. Profit when producing 80 items =  
4. Can you explain, from our model, why the company makes less profit when producing 10 more units?

Answer 1

P(x) = -0.5(x-120)^2 + 7200 - 50x -200

= -0.5x^2 +120x - 7200+7200 -50x -200

= -0.5x^2 +70x -200

domain is [0,140]

P(70) = 2250

P(80) = 2200

profit maximizing output level is found by taking the derivative and setting it =0

P'(x) = -x+70 = 0, x=70

the profit function is a parabola, downward opening with vertex or maximum at the point (70,2250)

at any other output level the profit is less. That's a math explanation. In economic terms though, the revenue function appears to reflect a company that's so large, with such a large market share that increasing output causes a drop in price, which causes reduced profits at high enough output levels. It's a monopoly.