The rate at which people enter an amusement park on a

Chapter 7, Problem 55

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The rate at which people enter an amusement park on a given day is modeled by the function E defined by E(t) !&t 2 " 1 2 5 4 6 t 0 ' 0&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) !&t 2 " 3 9 8 8 t 9 ' 0 &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 for 9 $ t $ 23, which are the hours that 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 P.M. (t ! 17)? Round your answer to the nearest whole number. (b) The price of admission to the park is $15 until 5:00 P.M. (t ! 17). After 5:00 P.M., the price of admission to the park is $11. How many dollars are collected from admissions to the park on the given day? Round your answer to the nearest whole number. (c) Let H(t) ! ! t 9 (E(x) " L(x))dx for 9 $ t $ 23. The value of H(17) to the nearest whole number is 3725. Find the value of H((17) and explain the meaning of H(17) and H((17) in the context of the park. (d) At what time t, for 9 $ t $ 23, does the model predict that the number of people in the park is a maximum?