The profit function of the school is given by:
P(t) = -60t^2 + 420t - 440
To find the price per ticket that maximizes profit, we need to find the vertex of the parabola that represents the profit function. The vertex of a parabola in standard form, y = ax^2 + bx + c, is given by:
x = -b/2a
In this case, a = -60 and b = 420, so:
x = -420/(2*(-60)) = 3.5
Therefore, the price per ticket that maximizes profit is $3.50 per ticket.
To find the appropriate range of prices, we need to determine where the profit function is increasing and where it is decreasing. We can do this by finding the critical points of the function, which are the values of t that make the derivative of the function equal to zero:
P'(t) = -120t + 420
-120t + 420 = 0
t = 3.5
This critical point corresponds to the vertex of the parabola, where the function changes from decreasing to increasing. Therefore, the range of appropriate prices is from $0 up to $3.50 per ticket and from $3.50 per ticket and up.