Math CentralQuandaries & Queries


Question from Kristy, a student:

A skating rink manager finds that revenue R based on an hourly fee x for skating is represented by the function R(x) = -200x^2 + 1500x

What is the maximum revenue and what hourly fee will produce

Hi Kristy,

I don't know if this is a problem from an algebra class or a calculus class. If this is a calculus problem then differentiate R(x) with respect to x and solve R'(x) = 0. This will give you one critical point. Use the second derivative test to verify thatthis critical point is in fact a maximum.

If this is an algebra problem then you know, from the form of the function R(x) = -200x2 + 1500, that the graph is a parabola. Also since the coefficient of x2 is negative the parabola opens downward. Solve R(x) = 0 to find the points where the graph crosses the x-axis. Since the axis of symmetry passes through the vertex of a parabola this parabolas reaches its maximum at an x-value that is half way between the points where the graph crosses the x-axis.

I hope this helps,

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