The rate at which people enter an amusement park on a given day is modeled by the function E defined by E(t) = 15600/(t^2 - 24t + 160). The rate at which people leave the same amusement park on the same day is modeled by the function L defined by L(t) = 9890/(t^2 - 38t + 370). Both E(t) and L(t) are measured in people per hour and time t is measured in hours after midnight. These functions are valid from 9 less than or equal to t less than or equal to 23, the hours during which the park is open. At time t = 9, there are no people in the park.
(a) How many people have entered the park by 5:00 PM (t = 17)? Round your answer to the nearest whole number.
(b) The price of admission to the park is $15 unti 5:00PM (t = 17). After 5:00PM, the price of admission to the park is $11. How many dollars are collected from the admissions to the park on the given day? Round your answer to the nearest whole number.
(c) Let H(t) = the integral from 9 to t E(x) - L(x) dx for 9 less than or equal to t less than or equal to 23. The value of H(17) to the nearest whole number is 3725. Find the value of H^1(17), and explain hte meaning of H(17) and H^1(17) in the context of the amusement park.
(d) At what time t, for 9 less than or equal to t less than or equal to 23, does the model predict that the number of people in the park is a maximum?