It should have this form:

P(t) = c + a sin(bt + d)

c = mid point = (200 + 40)/2 = 120

a = amplitude = (200 - 40)/2 = 80

b = angular frequency = 2πf, where f = linear frequency

f = 2 cycles per hour ⇒ b = 2πf = 2π(2) = 4π radians/hour

d = initial phase

So far we have

P(t) = 120 + 80sin(4πt + d), with t in hours, P(t) in lbs/ft^{2}

P(0) = 40 ⇒ 120 + 80sin[4π(0) + d] = 120 + 80sin(d)

40 = 120 + 80sin(d)

sin(d) = (40 - 120)/80 = -80/80 = -1

d = sin^{-1}(-1) = 3π/2 [in radians]

Finally,

**P(t) = 120 + 80sin(4πt + 3π/2), with t in HOURS, the argument of the sine in radians**, and P in lbs/ft^{2}

Test it:

Is P(0) = 40?

P(0) = 120 + 80sin(3π/2) = 120 + 80(-1) = 40 [good]

Is P_{max} = 200?

P_{max} occurs when the sine is maximum, i.e. when the sine is 1.

P_{max} = 120 + 80(1) = 200 [good]

Is P_{min} = 40?

P_{min} occurs when the sine is at its minimum of -1.

P_{min} = 120 + 80(-1) = 40 [good]

Is the linear frequency 2 cycles per hour?

f = 4π/(2π) cycles per hour = 2 cycles per hour [note the units are cycles per hour, not cycles per second as in Hz]

If you want the time in minutes, make the following change:

Replace 4π radians/hour with 4π/60 radians per minute = π/15 radians/minute. Then

**P(t) = 120 + 80sin[(π/15)t + 3π/2], with t in MINUTES.**

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