Question

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).

E(t) is measured in people per hours and time is measured in hours after midnight. The function is valid for9≤t≤ 23, the hours during which the park is open.
How many people enter the park from 9AM to 5PM (t=17) ?

Answer 1

Answer: 6004

Explanation:

`\int^(17)_(9) E(t) \text{ } dt=\int^(17)_(9) (15600)/(t^2 -24t+160)=15600 \int^(17)_(9) (1)/(t^2 -24t+160) \text{ } dt\\\\=15600\int^(17)_(9) (1)/((t-12)^2 +4^2) \text{ } dt\\ \\ =15600\left[(\arctan \left((t-12)/(4) \right))/(4) \right]^(17)_(9)\\\\=3900\left[\arctan \left((t-12)/(4) \right) \right]^(17)_(9)\\\\=3900\left(\arctan(5/4)-\arctan(-3/4))\\\\\approx 6004`