Question 289772
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 = {{{(-23)/(2*-4)}}}
x = {{{(-23)/(-8)}}}
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