Final answer:
The temperature can be modeled as a sinusoidal function with an equation of the form d(t) = A_*sin(B(t-C))+D. Given the temperature at different times and certain properties of sinusoidal functions, we can determine the values for the amplitude, period, phase shift, and vertical shift. Using the given information, we can find the equation for the temperature as a function of time.
Step-by-step explanation:
The temperature can be modeled as a sinusoidal function. In this case, the temperature is represented by the equation d(t) = A_*sin(B(t-C))+D. A is the amplitude, B determines the period, C represents the phase shift, and D is the vertical shift. Given that the temperature is 80 degrees at midnight, the high temperature is 88 degrees, and the low temperature is 72 degrees, we can determine the values for A, B, C, and D.
To find the amplitude, we subtract the low temperature from the high temperature and divide it by 2: A = (88-72)/2 = 8.
To find the period, we use the fact that the temperature goes through a complete cycle every 24 hours: B = 2π/24 = π/12.
Since the temperature is highest at noon, which is 12 hours after midnight, the phase shift is C = 12. Finally, the vertical shift is D = average of the high and low temperatures = (88+72)/2 = 80.
Therefore, the equation for the temperature, d, in terms of t is d(t) = 8*sin(π/12(t-12))+80.