Question

Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the high temperature for the day is 85 degrees and the low temperature of 45 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.

Answer 1

D(t) = 20 * sin(B * t - B * 3 + k * 2π) + 45 = 20 * sin(B * (t - 3) + k * 2π) + 45

Explanation:

The temperature over a day can be modeled by a sinusoidal function of the form:

D(t) = A * sin(Bt + C) + D

Where A is the amplitude (half the difference between the high and low temperatures), B is the angular frequency, C is the phase shift, and D is the vertical shift.

From the given information, we know:

The high temperature is 85 degrees.

The low temperature of 45 degrees occurs at 3 AM, or 3 hours after midnight.

So, the amplitude is (85 - 45)/2 = 20

The temperature at 3 AM can be found by setting t = 3:

D(3) = 20 * sin(B * 3 + C) + 45

So, the vertical shift is 45.

Finally, we can find the phase shift by using the fact that the low temperature occurs at 3 AM:

C = -B * 3 + k * 2π

Where k is an integer.

Putting it all together, we get:

D(t) = 20 * sin(B * t - B * 3 + k * 2π) + 45 = 20 * sin(B * (t - 3) + k * 2π) + 45

This is the equation for the temperature, D, in terms of t.