The manager of Three Fountains finds that revemue R based on an hourly fee F for skating is represented by the function R = -480F^2 + 5760F. What hourly fee will produce the maximum revenue?
To find the hourly fee that will produce the maximum revenue, we need to find the value of F that maximizes the function R = -480F^2 + 5760F.
To find the maximum of a quadratic function, we can use the formula F = -b/2a, where a and b are the coefficients in the quadratic equation.
In this case, a = -480 and b = 5760.
Using the formula, we have:
F = -5760 / (2 * -480)
F = -5760 / -960
F = 6
Therefore, an hourly fee of $6 will produce the maximum revenue.
To find the hourly fee that will produce the maximum revenue, we need to determine the maximum value of the revenue function.
The given revenue function is R = -480F^2 + 5760F.
To find the maximum value, we can use calculus by taking the first derivative of the function and setting it equal to zero.
Let's start by finding the derivative of the revenue function R with respect to F:
dR/dF = -960F + 5760
Setting this derivative equal to zero, we have:
-960F + 5760 = 0
Solving for F, we get:
-960F = -5760
F = -5760 / (-960)
F = 6
Therefore, an hourly fee of $6 will produce the maximum revenue for Three Fountains.
To find the hourly fee that will produce the maximum revenue, we need to determine the value of F when the revenue function R is at its maximum.
The revenue function is given as R = -480F^2 + 5760F.
To find the maximum, we need to find the vertex of the parabola defined by the revenue function. The x-coordinate of the vertex for a quadratic equation in the form y = ax^2 + bx + c is given by x = -b / (2a).
In this case, a = -480, b = 5760.
The x-coordinate of the vertex is calculated as follows:
F = -b / (2a) = -5760 / (2*(-480)) = 5760 / 960 = 6.
Therefore, the hourly fee that will produce the maximum revenue is $6.