Math Polynomial with steps

Question

For a certain company, the cost for producing x items is 40x+300 and the revenue for selling x items is 80x−0.5x2 

 

The profit that the company makes is how much it takes in (revenue) minus how much it spends (cost). In economic models, one typically assumes that a company wants to maximize its profit, or at least wants to make a profit! 

  

Part a: Set up an expression for the profit from producing and selling x items. We assume that the company sells all of the items that it produces. (Hint: it is a quadratic polynomial.) 

  

Part b: Find two values of x that will create a profit of $300. 

  

The field below accepts a list of numbers or formulas separated by semicolons (e.g. 2;4;6 or x+1; x−1). The order of the list does not matter. To enter a−−√, type sqrt(a). 

  

x=   

  

Part c: Is it possible for the company to make a profit of $15,000? 

Answer 1

Profit = 80x-.5x^2 - (40x+300)

= -x^2/2 +40x -300

Profit = 300 = -x^2/2 +40x -300

x^2/2 -40x +600 = 0

x^2 -80x +1200 = 0

x^2 -80x + 1600 = 1600 -1200

(x-40)^2 = 400

x -40 = +/- 20

x = 40+/-20

x = 20 or 60

maximum profit is when

P'(x) = -x + 40 = 0

x = 40

P(40) = 40(40) -.5(40^2) -300

= 1600 -800 -300 = $500

$15,000 profit is possible for a time period 30 times the time period when profit is $500