Final answer:
The sinusoidal function to model the daily outside temperature based on the given information is T(t) = 11 sin((2π/24)(t - 17)) + 85, where t is the number of hours since midnight.
Step-by-step explanation:
The student's question relates to modeling the daily outside temperature as a sinusoidal function in mathematics, specifically trigonometry. Given that the high temperature of 96 degrees occurs at 5 PM (17 hours after midnight), and the average temperature for the day is 85 degrees, we can define the sinusoidal function for temperature T(t) as:
T(t) = A sin(B(t - C)) + D
where:
- A is the amplitude (half the difference between the maximum and minimum temperatures),
- B is related to the period (how long it takes for one full cycle of temperature changes, which is typically 24 hours for a daily cycle),
- C is the horizontal shift (which tells us the time at which the maximum temperature occurs, 17 in this case), and,
- D is the vertical shift, which represents the average temperature for the day (85 degrees).
Since the high temperature is 11 degrees above the average, the amplitude A is 11. The period of the sinusoidal function is 24 hours, so B is (2π)/24. The function reaches its peak at 5 PM, so C is 17. Lastly, D is the average temperature, which is 85 degrees.
Hence the equation may look like this:
T(t) = 11 sin((2π/24)(t - 17)) + 85