Question 416950
Assume t=0 is 6:00 pm when the concentration is at a maximum.
12 hours later at 6:00 am, the concentration reaches a minimum.
The period of a sinusoidal function is the time required to complete one full cycle.  
So the period is 24 hours since 12 hours later at 6:00 pm the concentration will again be at a maximum.
The concentration ranges from 10 to 30, so the function needs to oscillate about the midpoint (average), which is 20.  
The maximum amplitude of these oscillations is +/- 10, since we have a maximum of 30 and a minimum of 10.
The general form of our periodic function will be {{{C(t) = C0*cos((2*pi/T)t)+C1}}} 
where C0 = the amplitude of oscillation and C1 is average value.
So our equation becomes {{{C(t) = 10*cos((pi/12)*t) + 20}}}
This equation fits the observations:
At t = 0 (6:00 pm) the concentration is {{{C(0) = 10*cos(0) + 20 = 30}}}
At t = 12 hrs (6:00 am) the concentration is {{{C(12) = 10*cos(pi) + 20 = -10 + 20 = 10}}}
And at t = 24 hrs (one full period) we are back to the maximum value of 30.
The function looks like this:

{{{graph(300,200,-24,24,-30,30,10*cos(pi*x/12)+20)}}},