Expert: Ahmed Salami - 5/6/2011

Question
A biological variable z has period of 24 hours and varies sinusoidally between the values 5 and 9. Taking t=0, to coincide with midnight, and measuring t in hours , find a formula expressing z as a function of t, given that z=5 at 4 p.m.

Answer
Hi Guneet,
The general sinusoidal expression is of the form;
z = Asin(wt + φ) + D
where A is the amplitude and φ is the phase difference which shows where in the cycle the oscillations begin at t = 0.
w is the angular frequency and is related to the period T by the formula;
w = 2π/T = π/12
D is a constant which shows how much the waveform has been displaced vertically, which is the average of the maximum and minimum values.
D = (5 + 9)/2 = 7
Also, the difference between the maximum and minimum values is equal to 2A.
2A = 9 - 5 = 4
A = 2
Now, at t = 16 (equivalent to 4 pm), z = 5 and we have
5 = 2sin(16π/12 + φ) + 7
-2 = 2sin(4π/3 + φ)
sin(4π/3 + φ) = -1
4π/3 + φ = arcsin(-1) = 3π/2
φ = π/6

Therefore,
z = 2sin(πt/12 + π/6) + 7
= 2sin[π(t/2 + 1)/6] + 7

Regards