The temperature in a certain cave is a sinusoidal function of time. When you first enter the cave, you set up a thermometer to record the temperature. After 19.5 hours, the temperature in the cave is at its maximum: 35 degrees celsius. The temperature then dropped, reaching the minimum temperature of 25 degrees Celesius in 36.5 hours after you entered the cave.

During the first 90 hours after you enter the cave, how much time will the temperature in the cave be above 31 degrees Celsius?

I need help finding the period and the phase shift:
sinusoidal model
T(t)=sin[2pi/B(x-C)]+D

D=(6+5)/2=11/2
A=(6-5)/2=0.5
B=period
C=phase shift

I need help finding B and C

Since the max-to-min temperature interval is 17 h, the period is 34 h. Use the time of the maximum to get the phase angle. The argument of the sin function is then pi/2.

To find the period (B) and the phase shift (C) in the sinusoidal model, we can use the information given about the temperature in the cave.

For a sinusoidal function, the period is the time it takes for the function to complete one full cycle. In this case, the temperature reaches its maximum and minimum values in 19.5 hours and 36.5 hours, respectively.

The period of the sinusoidal function is the difference between the times when the temperature reaches its maximum and minimum values:
B = 36.5 hours - 19.5 hours
B = 17 hours

So the period of the function is 17 hours.

Next, let's find the phase shift (C). The phase shift represents the horizontal shift of the sinusoidal function. It tells us where the function starts or ends.

In this case, the temperature reaches its maximum value after 19.5 hours, which means the function starts at that point. The standard form of the sinusoidal model is T(t) = sin[2pi/B(t - C)] + D.

Since the temperature reaches its maximum value after 19.5 hours, the phase shift is equal to the time it takes for the function to start:
C = 19.5 hours

So the phase shift of the function is 19.5 hours.

Therefore, the sinusoidal model for the temperature in the cave is:
T(t) = sin[2pi/17(t - 19.5)] + 11/2

Now, let's find how much time the temperature in the cave will be above 31 degrees Celsius during the first 90 hours after you entered the cave.

We can evaluate the temperature function for each hour between 0 and 90 and count the number of times the temperature is above 31 degrees Celsius.

Plug in the values of T(t) and evaluate for each hour between 0 and 90:
T(t) = sin[2pi/17(t - 19.5)] + 11/2

Count the number of times T(t) is above 31 degrees Celsius for each hour and sum them up to find the total time.