Final answer:
The function for the temperature, D, in terms of t (the number of hours since noon) is modeled using a sinusoidal function as D(t) = 10sin(2πt/24)+80, where the amplitude is 10 degrees and the period of the temperature cycle is 24 hours.
Step-by-step explanation:
To find a function for the temperature, D, in terms of t (the number of hours since noon), we should consider how temperature typically changes throughout the day. Because the question doesn't specify the type of function or the model of the temperature change, we might assume a sinusoidal pattern that is typical for daily temperature, which rises to a high and falls to a low in a roughly predictable way.
Let us start by setting up the midline of the function. Since the high is 90 degrees and the low is 70 degrees, the midline will be at the average of these two temperatures, which is (90+70)/2 = 80 degrees. This matches with the given temperature of 80 degrees at noon, so no vertical shift is necessary.
The amplitude of the function is half the difference between the high and low temperatures. So, the amplitude will be (90-70)/2 = 10 degrees.
Without knowing the specific times when the high and low occur, we could presume the simplest case: the high occurs once every 12 hours and the low occurs once every 12 hours, which implies a full cycle every 24 hours. This gives us a period of 24 hours for the sinusoidal function. In general, a sine function has a period of 2π, so the angular frequency, ω, would be 2π/24.
We can now establish the sine function for the temperature, D(t), to be:
D(t) = 10sin(2πt/24)+80. Here, t is measured in hours since noon.