Expert: Alon Mandes - 5/24/2010

Question
A population of animals oscillates sinusoidally between a low of 100 on January 1 and a high of 1200 on July 1. Graph the population against time and use your graph to find a formula for the population  as a function of time , in months since the start of the year.

Answer
The general form of a sinusoidal relation is given by y(t)=A+B*Sin(C*t+D) . All we need to
do is to find A,B,C & D . To do so, we use the information given above :
1. Low value (minimum) of the function is 100 reached at  January 1 (t=1) .
2. High value(maximum) of the function is 1200 reached at July 1  (t=7) .
3. The function is periodic , that means y(1)=y(13)
Minimum is reached when sin(x)=-1 & maximum is reached when sin(x)=1 , thus we can write :
1. A-B=100
2. A+B=1200
Solving this linear system of equation gives us : A=650 , B=550 .
Now, our function becomes : y(t)=650+550*Sin(C*t+D) .
To find C & D , we use the points of time where the formula reached it's max & min :
At t=1 we have min --> 100=650+550*Sin(C+D)  --> Sin(C+D)=-1 --> C+D=π/2
At t=7 we have max --> 1200=650+550*Sin(7C+D)--> Sin(7C+D)=1 --> 7C+D=3π/2
Solving this linear system of equation yields : c=π/6 , D=π/3 .

So, our function is :
y(t)=650+550*Sin{ [π(t+2)/6] }

Alon.