Application of a sinusoidal function

Question

Suppose a sales can be modelled by the equation;

 

S=200+200cos[pi/6(t+2)]

 

where t is time in months with t=0 corresponding to January for what month are sales equal to 0?

Answer 1

So lets find the values of t where S=0.  Note we need to restrict the possible solutions for t to the interval [0,11] (ie January through December).

 

This resolves into an exercise of solving a trigonometric function:

 

0=200+200cos[(t+2)pi/6)

-200=200cos[(t+2)pi/6]

-1=cos[(t+2)pi/6]

 

Taking the inverse cosine of both sides

 

cos^-1(-1)=cos^-1(cos[(t+2)pi/6])=(t+2)pi/6

 

Note that cos^-1(-1) has a solution at pi and every 2pi after that...

 

(t+2)pi/6=pi+N2pi where N=0,+/-1,+/-2,etc...

 

Now lets solve for t

 

t+2=6+N12pi

t=4+N12pi

 

So now we need to find the values of N where t is between 0 and 11

 

for N=0 we have

 

t=4+0*12pi=4

 

for all other values of N, t falls outside the interval [0,11].

 

Since N=4 equates to May, the answer is May.

 

Good Luck

 

Cheers

 

-Chris