Final answer:
The equation for temperature D as a function of time t is D(t) = 12 · sin(π/12(t - 6)) + 80. This sinusoidal function takes into account the high and low temperatures of the day, and the time at which the low temperature occurs.
Step-by-step explanation:
To find the equation for the temperature D in terms of t (the number of hours since midnight), given the high and low temperatures for the day and the time of the low temperature, we can use a sinusoidal function. The general form for a sinusoidal function is:
D(t) = A · sin(B(t - C)) + D
Where:
- A is the amplitude of the wave,
- B is the frequency,
- C is the horizontal shift (phase shift),
- D is the vertical shift (midline of the wave).
The amplitude A is half the difference between the high and low temperatures:
A = (92 - 68) / 2 = 12 degrees
The vertical shift D is the average temperature, which is the sum of the high and low temperatures divided by 2:
D = (92 + 68) / 2 = 80 degrees
Assuming that the high temperature occurs at midday (approximately 12 PM or 12 hours after midnight), and the low occurs at 6 AM (6 hours since midnight), the function will have a period (P) of 24 hours. The frequency B is calculated by B = 2π / P, and given 24 hours, B equals π/12.
The horizontal shift C is equal to the time of the low temperature, which is 6 hours after midnight:
C = 6 hours
The resulting sinusoidal function for temperature D as a function of t is:
D(t) = 12 · sin(π/12(t - 6)) + 80