1. 5:00 PM is 17 hours after midnight. 9:00 AM is 9 hours after midnight.
Part (a)
The number of people who’ve entered the park can be found by integrating E(t)--the rate at which people enter the park.
5:00 PM is 17 hours after midnight.
9:00 AM is 9 hours after midnight.
15600
t2-24t+160 dt 9 17 =
6004 people will enter the park between 9:00 AM and 5:00 PM.

2. We’ll multiply this result by the full-day price. ($15)
Part (b)
To find how much money the park has collected, we need to integrate E(t) two different times.
We’ll multiply this result by the full-day price. ($15)
15600
t2-24t+160 dt 9 17 23
And this gets multiplied by the evening price. ($11)

3. Part (b) 15600 dt = (15)(6004.270) (15) t2-24t+160 (11)9 17 23
(15)
= (15)( )
(11)
= (11)( )

4. The park will take in $104,048 on a typical day.
Part (b)
(15)( )
104,
(11)( )
The park will take in $104,048 on a typical day.

5. Part (c)
H(t) would represent the number of people in the park at any given time.
Since H(17) = 3725, that’s how many people were in the park at 5:00 PM.
H’(17) represents the rate at which people are entering/exiting the park at 5:00 PM.

6. Part (c)
H’(t) = E(t) – L(t)
H’(17) = E(17) – L(17)
H’(17) = –
H’(17) =
H’(17) represents the rate at which people are entering/exiting the park at 5:00 PM.
This means that at 5:00 PM, the park’s attendance is dropping at a rate of 380 people/hour.

7. The number of people in the park, H(t), will be a max when H’(t) = 0.
Part (d)
The number of people in the park, H(t), will be a max when H’(t) = 0.
H’(t) = E(t) – L(t)
If H’(t) = 0, then E(t) = L(t).
15600
t2-24t+160
9890
t2-38t+370 =

8. Part (d)
The algebra involved here would be absolutely insane to attempt. Instead, graph both functions and find where they intersect.
t = or
(3:48 PM)
15600
9890 =
t2-24t+160
t2-38t+370