Expert: Paul Klarreich - 12/19/2008

Question
the population of a clownfish around a certain reef varies sinusoidally.  at a time 2.4 yrs into the study, the population was at its maximum of 6000.  a low point of 2000 was occurred 5.6 yrs into the study.  Question:  1.  what is the first two times that the number of clownfish was 3000.  Question 2  Estimate the population at the 3 yr mark?

Answer
Questioner:   cheryl johnson
Category:  Calculus
Private:  No
 
Subject:  pre calc
Question:  the population of a clownfish around a certain reef varies sinusoidally.  at a time 2.4 yrs into the study, the population was at its maximum of 6000.  a low point of 2000 was occurred 5.6 yrs into the study.  Question:  1.  what is the first two times that the number of clownfish was 3000.  Question 2  Estimate the population at the 3 yr mark?
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Hi, Sheryl,

You are looking for a function having the form:

P(t) = Amplitude * sin(At + B) + Offset

given these facts:
....................
at a time 2.4 yrs into the study, the population was at its maximum

-- conclusion:

t = 2.4 gives an 'argument' (angle?) equal to pi/2

A(2.4) + B = pi/2
.....................
a low point occurred 5.6 yrs into the study

t = 5.6 gives an 'argument' (angle?) equal to 3pi/2

A(5.6) + B = 3pi/2

I think you can combine those two facts to find A and B.


A(3.2) = pi;  A = pi/3.2

Subst:

pi/3.2(5.6) + B = pi/2

You can do that arithmetic.

..........................

Now,

IF   P(t) = Amplitude sin(At + B) + Offset


Then  Pmax = Amplitude + offset
while Pmin = -Amplitude + offset

AND the population was at its maximum of 6000  

-- means Pmax = 6000

a low point of 2000

-- means Pmin = 2000.

You should be able to use those two facts to find Amplitude and Offset, and thus write your function completely.  I think you'll find Amplitude = 2000, Offset = 4000.

Then you can answer the questions:


1.  what is the first two times that the number of clownfish was 3000.

Solve P(t) = 3000.  You will want to use your knowledge of basic right-triangle trigonometry.

2. Estimate the population at the 3 yr mark?

This is just P(3).
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If you still have trouble, let me know.  But when you do, please:

1. PROOFREAD.

2. ORGANIZE.  Would you like to read my answer like this?

You are looking for a function having the form: P(t) = Amplitude * sin(At + B) + Offset  given these facts: at a time 2.4 yrs into the study, the population was at its maximum : t = 2.4 gives an 'argument' (angle?) equal to pi/2.  A(2.4) + B = pi/2. a low point occurred 5.6 yrs into the study.  t = 5.6 gives an 'argument' (angle?) equal to 3pi/2.  A(5.6) + B = 3pi/2  I think you can combine those two facts to find A and B.  A(3.2) = pi;  A = pi/3.2
Substitute: pi/3.2(5.6) + B = pi/2.   You can do that arithmetic.