SOLUTION: Hello can anyone help solve this problem with work shown and answers it will help me learn how to do these problems for my upcoming test. Sean's Widget Company has revenue tha

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: Hello can anyone help solve this problem with work shown and answers it will help me learn how to do these problems for my upcoming test. Sean's Widget Company has revenue tha      Log On


   

Question 289772: Hello can anyone help solve this problem with work shown and answers it will help me learn how to do these problems for my upcoming test.

Sean's Widget Company has revenue that is modeled by the function R(x) = -4x2 + 35x and costs modeled by the function C(x) = 12x +30. The input variable x is in hundreds of widgets, and the outputs of the functions are in units of thousands of dollars. Construct a profit function P(x) and find the value of x that maximizes the profit.

Thank you John
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
Sean's Widget Company has revenue that is modeled by the
function R(x) = -4x2 + 35x and costs modeled by the function C(x) = 12x +30.
The input variable x is in hundreds of widgets, and the outputs of the
functions are in units of thousands of dollars.
Construct a profit function P(x) and find the value of x that maximizes the profit.
:
Profit = Revenue - Cost
P(x) = R(x) - C(x)
Which is
P(x) = -4x^2 + 35x - (12x + 30)
Removing the bracket changes the signs
P(x) = -4x^2 + 35x - 12x - 30
Combine like terms
P(x) = -4x^2 + 23x - 30
:
A quadratic equation, we can find the axis of symmetry: x = -b/(2a)
That value of x will give us max profit
x = %28-23%29%2F%282%2A-4%29
x = %28-23%29%2F%28-8%29
x = +2.875 hundreds of widgets
:
287.5 ~ 288 widgets for max profit (has to be an integer)
:
:
To find the actual profit value (in thousands), substitute 2.88 for x in the
equation P(x) = -4(2.88^2) + 23(2.88) - 30