Please help to solve this one

Question

Outside temperature over a day can be modelled as a sinusoidal function. Suppose you know the high temperature for the day is 52 degrees and the low temperature of 38 degrees occurs at 3 AM. Assuming t is the number of hours since midnight, find an equation for the temperature, D, in terms of t.

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Answer 1

If the max is 52 and the min is 38, then the sinusoidal oscillation will occur around the middle (average) of those temperatures. (52 + 38)/2 = 45.

Every day the oscillation cycle repeats, meaning the period is 24 hours.

Since the low temperature occurs at 3 a.m., 3 hours after midnight, and then begins to increase, the graph would look like this:

We'll let the "x" variable be time (t) and we'll let the "y" variable be Temperature (D). So we will have a function of D(t) - Temperature as a function of time.

Since the max temp is near the origin, I'll chose a cosine function to model this. But it will be a negative cosine since it is flipped from the normal cosine.

So the generic function will be:

D(t) = -Acos(B(t - C)) + D

The horizontal shift is 3 hours to the right C = 3

The vertical shift will be +45 to move the graph up from zero to 45. So D = 45.

The amplitude is 7 because the temps are 45 ± 7 degrees so A = 7.

Because the period is 24 hours and the period = 2π/B then B = π/12

That makes the function:

D(t) = -7cos(π/12(t - 3)) + 45