Environmental Fluctuations Problem 4
Tutorial to help us answer problem 4.
The function
oscillates between 72 - 19 = 53 °F and 72 + 19 = 91 °F.
For cosine functions, all functional values, except the maximum and minimum, are realized twice in one period. The maximum and minimum values are only realized once in one period. This has to be true based on the fact that the function will maximize and minimize and come back to the starting functional value in one period. In our problem, we know that the period is 24 hours and we found the maximum and minimum values of the function are 53 and 91, respectively. This means that over any 24 hour period, 53 and 91 degrees F will be realized once while all other temperatures strictly between 53 and 91 will be realized twice. Since 83.1 is strictly between 53 and 91, 83.1 degrees F will be realized twice over any 24 hour period of time.
Looking at the graphs of sine and cosine functions below, we see that all functional values satisfying -1 < y < 1 are attained twice in a single period.
