Final answer:
To find the temperature at 7 AM, a sinusoidal function is needed with given data of high and average temperatures. However, complete details such as the low temperature and its occurrence time are essential for an accurate calculation.
Step-by-step explanation:
The question pertains to modeling the daily temperature variation as a sinusoidal function and determining the temperature at a specific time, using given maximum temperature and average temperature data points. With a high of 84°F occurring at 6 PM and an average temperature of 70°F, we need to use the properties of sinusoidal functions to find the temperature at 7 AM.
The sinusoidal function that models temperature could be expressed in the form:
T(t) = A cos(B(t - C)) + D
Where:
- A is the amplitude
- B relates to the period of the function
- C is the horizontal shift (phase shift)
- D is the vertical shift (average temperature)
Based on the given information, D is 70°F, the vertical shift since that is the average temperature. The amplitude, A, is calculated by finding the difference between the high temperature and the average temperature. Hence, A = 84°F - 70°F = 14°F. The period of the function is 24 hours because the temperature cycle completes once each day. However, we do not have enough information to calculate B or C, which is necessary for an exact value at 7 AM. To estimate the temperature at 7 AM, additional details are needed, such as the low temperature and its time of occurrence.