Question

One fine day in Rochester the low temperature occurs at 5 a.m. and the high temperature at 5 p.m. The temperature varies sinusoidally all day. The temperature t hours after midnight isT(t) = A + B sin left( frac{pi (t-C)}{12} right) where A , B , and C are certain constants. The low temperature is 26 and the high temperature is 34. Find the average temperature during the first 2 hours after noon. Hint: The high and low temperatures can be used together to find A and B. Determine C from the fact that it is hottest at 5 p.m.

Answer 1

To find the average temperature between noon and 2 p.m. in Rochester from a sinusoidal function representing daily temperature change, we use the high and low temperature values to calculate the constants for the sinusoidal function and then integrate over the desired time period.

Step-by-step explanation:

The question asks us to find the average temperature during the first 2 hours after noon, given a sinusoidal function that models the temperature throughout the day in Rochester. The function provided is T(t) = A + B sin(&frac;{pi (t-C)}{12}), where A, B, and C are constants, with the low temperature being 26 at 5 a.m. and the high temperature being 34 at 5 p.m.

Since we know the high and low temperatures, we can determine constants A and B. The average of the high and low temperatures is (26 + 34) / 2 = 30, which is A, the midpoint of the sinusoidal function. Therefore, B, the amplitude, is half the difference between the high and low temperatures: (34 - 26) / 2 = 4. The constant C represents the time shift to match the peak temperature which occurs at 5 p.m., so C is 5 (since the hottest point is 17 hours after midnight).

To find the average temperature between 12 p.m. and 2 p.m., we evaluate the integral of the temperature function from t=12 to t=14 and then divide by the interval length, which is 2 hours.