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Topic: [ap-calculus] logistic question
Replies: 1   Last Post: Oct 6, 2008 8:47 PM

 Stu Schwartz Posts: 308 Registered: 8/23/06
Re: [ap-calculus] logistic question
Posted: Oct 6, 2008 8:47 PM

On Oct 6, 2008, at 6:00 PM, Louise Doornek wrote:

> I have taught AB Calc for 18 years, but am teaching BC for the
> first time this year. I am trying to do the BC practice exam that
> was published last spring on the AP Audit section of The College
> Board website. I am stuck on problem #84. Could someone please
> help me? Here is the problem.
>
> The rate of change, dP/dt, of the number of people on an ocean
> beach is modeled by a logistic differential equation. The maximum
> number of people allowed on the beach is 1200. At 10 a.m., the
> number of people on the beach is 200 and is increasing at the rate
> of 400 people per hour. Which of the following differential
> equations describes the situation? The correct answer is C.
>
> (A) dP/dt = (1/400)(1200-P)+ 200
> (B) dP/dt = (2/5)(1200-P)
> (C) dP/dt = (1/500)P(1200-P)
> (D) dP/dt = (1/400)P(1200-P)
> (E) dP/dt = 400P(1200-P)
>

The general equation for logistic growth is dP/dt = kP(C - P) where C
is the carrying capacity.

Your given information is: C = 1200, P = 200, dP/dt = 400. You need
to find k.

So,

dP/dt = kP(C - P)
400 = k(200)(1200 - 200)
400 = 200000 k
k = 1/500

Therefore: dP/dt = (1/500)P(1200 - P)

- Stu Schwartz

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Course related websites:
http://apcentral.collegeboard.com/calculusab
http://apcentral.collegeboard.com/calculusbc
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