Question 1071397
c = 50x + 600
r = 100x - .5x^2


p = r - c = 100x - .5x^2 - (50x + 600)


simplify to get:


p = 100x - .5x^2 - 50x - 600


combine like terms to get:


p = 50x - .5x^2 - 600


reorder the terms in descending order of degree to get:


p = -.5x^2 + 50x - 600


when p = 600, this equation becomes:


600 = -.5x^2 + 50x - 600


subtract 600 from both sides of this equation to get:


0 = .5x^2 + 50x - 1200


multiply both sides of this equation by -2 to get:


0 = x^2 - 100x + 2400


factor this equation to get:


(x-40) * (x-60) = 0


solve for x to get x = 40 and x = 60.


the profit will be 600 when x = 40 and when x = 600


to confirm, go back to the original equation of p = -.5x^2 + 50x - 600


when x = 40, this equation becomes:
p = -.5*40^2 + 50*40 - 600 which becomes:
p = -800 + 2000 - 600 which becomes:
p = 2000 - 1400 which becomes:
p = 600


when x = 60, this equation becomes:
p = -.5*60^2 + 50*60 - 600 which becomes:
p = -1800 + 3000 - 600 which becomes:
p = 3000 - 2400 which becomes:
p = 600


you can also solve this graphically by graphing y = -.5x^2 + 50x - 1200 and also graphing y = 600 and looking for the intersection points.


your graph will look like this:


<img src = "http://theo.x10hosting.com/2017/030404.jpg" alt="$$$" </>