Outside temperature over a day can be modelled as a sinusoidal function. Suppose you know the high temperature of 60 degrees occurs at 3 PM and the average temperature for the day is 45 degrees. Assuming t is the number of hours since midnight, find an equation for the temperature, D, in terms of t.

To find an equation for the temperature D in terms of t, we can use the information given.

We know that the high temperature of 60 degrees occurs at 3 PM, which is 15 hours since midnight. This will be the maximum value of our sinusoidal function.

We also know that the average temperature for the day is 45 degrees. Since the sinusoidal function is centered around the average temperature, the midline will be D = 45.

Let's assume the period of the sinusoidal function is 24 hours since we are considering a full day.

The general form of a sinusoidal function is given by: D = A sin(B(t - C)) + D

A is the amplitude, B is the frequency, C is the phase shift, and D is the midline.

In this case, we know the amplitude is (60 - 45) / 2 = 7.5 (half the difference between the maximum and average temperature).

The frequency can be calculated using the formula B = (2π) / period. So in this case, B = (2π) / 24 ≈ 0.26.

The phase shift C is the number of hours after midnight when the maximum temperature occurs. In this case, C = 15.

Substituting the values into the equation, we get:

D = 7.5 sin(0.26(t - 15)) + 45

Therefore, the equation for the temperature D in terms of t is:

D = 7.5 sin(0.26(t - 15)) + 45

To find an equation for the temperature, D, in terms of t, we can use the form of a sinusoidal function:

D = A * sin(B * (t - C)) + D_average

Where:
- A is the amplitude of the sinusoidal function (half the difference between the maximum and minimum values).
- B is the frequency of the sinusoidal function (2π divided by the period).
- C is the phase shift of the sinusoidal function (the horizontal shift).
- D_average is the average temperature for the day.

From the given information:
- The high temperature occurs at 3 PM, which is 15 hours since midnight. So, the phase shift, C, is 15.
- The average temperature, D_average, is 45 degrees.
- The difference between the maximum and minimum (amplitude) is 60 - 45 = 15. Therefore, the amplitude, A, is 15/2 = 7.5.
- The period of the function is 24 hours (full cycle).

Now, we can plug these values into the equation:

D = 7.5 * sin((2π/24) * (t - 15)) + 45

Therefore, the equation for the temperature, D, in terms of t is:

D = 7.5 * sin((π/12) * (t - 15)) + 45

Assume y = m + Asin(k(t-h))

center line at (60+45)/2 = 52.5
y = 52.5+sin(k(t-h))

amplitude is (60-45)/2 = 7.5
y = 52.5+7.5sin(k(t-h))

minimum is at t=15 which is 3PM, so
y = 52.5 - 7.5cos(k(t-15))

period is 24 hours, so 2π/k = 24
y = 52.5 - 7.5cos(π/12 (t-15))