SOLUTION: Given the following revenue and cost functions, find the x-value that makes profit a maximum. (Re2all that profit equals revenue minus cost.) R(x) = 55x - 2x^2; C(x) = 21x + 98

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: Given the following revenue and cost functions, find the x-value that makes profit a maximum. (Re2all that profit equals revenue minus cost.) R(x) = 55x - 2x^2; C(x) = 21x + 98       Log On


   

Question 613258: Given the following revenue and cost functions, find the x-value that
makes profit a maximum. (Re2all that profit equals revenue minus cost.)
R(x) = 55x - 2x^2; C(x) = 21x + 98
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Since profit = revenue - cost, let's make a profit function, P(x) that will be R(x) - C(x):
P%28x%29+=+R%28x%29+-+C%28x%29+=+%2855x+-+2x%5E2%29+-+%2821x+%2B+98%29
Note the use of parentheses! Without them we would not realize that both the 21 and the 98 should be subtracted.
P%28x%29+=+R%28x%29+-+C%28x%29+=+%2855x+-+2x%5E2%29+-+%2821x+%2B+98%29+=+-2x%5E2+%2B+34x+-98

Now that we have our profit function, we can see that:
  • Its graph will be a parabola because of the squared term.
  • The parabola will open downward because of the negative coefficient, -2, in front of the squared term.
  • The highest point (which would be the maximum profit) on the downward parabola would be the vertex of the parabola.
From the above we now know that we want to find the vertex of the profit parabola.

The x coordinate of the vertex of a parabola will be -b/2a where the "b" is the coefficient of the x term and the "a" is the coefficient of the x squared term. From
P%28x%29+=+-2x%5E2+%2B+34x+-98

we can see that your "a" is -2 and your "b" is 34. So the x coordinate of the vertex (which is where the maximum profit is) will be:
-34%2F%28-2%29+=+17