Thread: Sinusoidal functions, predicting temperature when given time?

Ok, we've been getting these problems lately and our teacher is making us review them and I did horrible on my last test because of a problem similar to this, I don't understand this at all, all I do know is that it involves moving the sine or cosine graph and anything after that our teacher was very vague about and let us wander on our own. If anyone can explain anything from this problem that would help, I'd be grateful

The temperature during the day can be approximated by a sinusoidal function. At 4 am, the temperature was a low at 65 fahrenheit, and then at 4 pm, the temperature hit a high of 103 fahrenheit

a) write an EQ which allows you to predict the temperature at t hours after midnight

b) find the temperature at 11 am

c) find the first time in the day when the temp reaches 98 fahrenheit

d) approximate the rate of change of the temperature at 2 pm

e) at what time is the rate of change of the temperature with respect to time the greatest?

As described, the curve is decreasing at t = 0 (that is, at midnight), vewry shortly reaching its minimum and then increasing to its maximum. It is up to you whether you use a sine curve shifted far to the left or a cosine curve shifted a little to the right.

If the high and low points of the curve are twelve hours apart, then what is the period (from high to low, and then back to high)?

If the maximum is 103 and the minimum is 65, then what is the amplitude? What is the vertical shift (so that this amplitude is "centered" correctly)?

If the minimum occurred at t = 4, then what is the phase shift, given the above period?

If you get stuck, please reply showing your work and reasoning so far. Thank you!