Question

In Seattle on September 30, the temperature hours after midnight was given by the function y=60 + 12sin((pi/x)(x-11))
What was the average temperature over the period from 8 A.M. until 10 P.M.?

Answer 1

The function does not exist at midnight (x=0)and is "weird" between 1am and 7am, as the temperature goes through many cycles. Eventually, it rises as the day goes on until mid-night.

In any case, to find the average from 8am to 10pm, we would integrate
f(x)=60 + 12sin((pi/x)(x-11))
with respect to x from x=8 to x=22, and divide by the time difference (22-8)=14.
The integral, however, is not easy to calculate because it involves gamma function.
If you have done numerical integration before (such as simpson's rule,etc.) you can get the answer using numerical integration to get 65.8 degrees.