Question 338218
This problem seems strange to me in that profit equations tend to be negative parabolas.  Therefore, the vertex of the parabola is when profit is maximized.  However, your equation is a positive parabola, which would lead me to believe there is some sort of error in the question.  The equation you wrote seems correct based on the info given, as well as your substituting 30 for x to find the profit, given 30 excursions.  However maximizing profit seems impossible, given that the parabola increases toward infinity.
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Ok, so the equation is {{{P= -0.4x^2+40x-800}}} So to maximize the profit you need to find the vertex of the parabola.  You can do that on a graphing calculator or by using the equation {{{x=-b/2a}}} where a=-0.4 and b=40.  So {{{x=-40/(2*-0.4)}}}
This equals 50. So at 50 excursions per month, the company maximizes profits.  If you want to know how much profit that is, substitute 50 for x in your profit equation.  {{{P=-0.4(50)^2+40(50)-800}}} P=$200