A skating rink manager finds that revenue R based on an hourly fee F for skating is represented by the function R= -480(F - 3.25)^2 + 5070. What hourly fee will produce maximum revenue?
To find the hourly fee that will produce maximum revenue, we need to determine the value of F that maximizes the function R.
The given revenue function is represented as R = -480(F - 3.25)^2 + 5070.
To find the maximum value of R, we can analyze the quadratic equation in the revenue function. The quadratic term in the function is (F - 3.25)^2.
Since the coefficient of the quadratic term is negative (-480), the parabola will be facing downwards. A downward-facing parabola with a negative quadratic coefficient has its maximum value at the vertex.
We can use the formula for the x-coordinate of the vertex of a quadratic function: x = -b / (2a), where a and b are the coefficients of the quadratic equation.
In this case, the coefficient of the quadratic term is a = -480, and there is no linear term, so b = 0. Hence, the formula for the x-coordinate of the vertex becomes: F = -0 / (2 * -480) = 0.
Substituting the value of F = 0 into the revenue function R = -480(F - 3.25)^2 + 5070, we can find the maximum revenue:
R = -480(0 - 3.25)^2 + 5070
R = -480(-3.25)^2 + 5070
R = -480(10.5625) + 5070
R = -5064 + 5070
R = 6
Therefore, when the hourly fee is F = $0, it will produce the maximum revenue of R = $6.
To find the hourly fee that will produce maximum revenue, we need to determine the value of F that maximizes the revenue function R.
The revenue function is given by R = -480(F - 3.25)^2 + 5070.
To find the maximum, we can observe that the quadratic term (F - 3.25)^2 will always be non-negative. Therefore, the maximum revenue occurs when this quadratic term is zero.
Setting (F - 3.25)^2 = 0, we solve for F:
F - 3.25 = 0
F = 3.25
Therefore, the hourly fee that will produce maximum revenue is $3.25.
Since you have no calculus (the easy way), lets do it this way. Obviously, it is a parabola going up, then down. So look for the half way point between both x intercepts.
So find the x intercepts.
0=-480(F-3.25)^2 + 5270
F^2+6.5F+3.25^2-5070/480=0 check that, then put it in the form
af^2 + bF+ C=0 and either factor it, or use the quadratic equation.
Then finally, take the average of the two roots, and you have the point of max revenue.