Expert: Paul Klarreich - 1/3/2010

Question
The rate at which people enter an amusement park on a given day is:
e(t) = 15600/ (t^2-24t+160)

The rate at which people leave the same park on the same day
is:
L(t)= 9890/(t^2-38t+370)

Both E(t) and L(t) are measured in people per day and time t is measured in
hours after midnight. These functions are defined for t ∈[9,23] . At t = 9 there
are no people in the park.
a) How many people have entered the park by 5pm (t = 17)?

b) the price of admission to teh part 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 you answer to teh nearest whole number

c) let h(t)= interval from 1 to 9 of the funciton (E(x)-L(x))dx for 9<equal t<equal 23. the value of H(17) to teh nearest whole number is 3725. Find the value of H'(17) and explain in teh meaning of H(17) and H'(17) in teh context of teh amusement park.

d)at what time t for 9<equal t<equal 23, does the model predict that the number of poeple in the park is a maximum?  

Answer
Questioner: Leah
Country: United States
Category: Calculus
Private: No
Subject: AP calc BC
Question: The rate at which people enter an amusement park on a given day is:
e(t) = 15600/ (t^2-24t+160)

The rate at which people leave the same park on the same day is:
L(t)= 9890/(t^2-38t+370)
...................
Both E(t) and L(t) are measured in people per day and time t is measured in
hours after midnight. These functions are defined for t ∈[9,23] . At t = 9 there are no people in the park.
............................
Is your E(t) the same as your e(t)?  I am going to assume you are simply careless and the answer is 'no'.

I am going to assume that  e(t) and l(t) are your rates.  E and L will mean something else.

[Don't do it again.]
..............................
Compute E(t) by integrating  e(t) (use partial fractions, I think) and using:

e(9) = 0

to find your constant of integration. (Your 'C')

and then:

Compute L(t) by integrating  l(t) and using:

l(9) = 0
................................


a) How many people have entered the park by 5pm (t = 17)?

This does not seem to be stated clearly, but:

This would be E(17)


b) the price of admission to teh part 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 you answer to teh nearest whole number

This looks like 15* E(17) + 11 *(E(23) - E(17))

c) let h(t)= interval from 1 to 9 of the function (E(x)-L(x))dx for 9<equal t<equal 23. the value of H(17) to teh nearest whole number is 3725. Find the value of H'(17) and explain in teh meaning of H(17) and H'(17) in teh context of teh amusement park.

Sorry -- this is incoherent.  Try it again.

d)at what time t for 9 <= t <= 23, does the model predict that the number of poeple in the park is a maximum?

This would be the max of L(t) - E(t), which should be the value of t where l(t) - e(t) = 0.