Find the average temperature

Question

In a certain city the temperature (in °F) t hours after 9 a.m. was modeled by the function

T(t)= 56+19sin(pi(t)/12)

Find the average temperature T_ave (in °F) during the period from 9 a.m. to 9 p.m.

Answer 1

Before looking at this particular problem, let's think about how to find the average value for a set of numbers, say [1, 3, 5, 2, 4]. To find the average of these five numbers, we add them up and divide by how many numbers we have: average=(1+3+5+2+4)/5=3. We will do the same thing with our temperature function.

To add up all of the values from 9am to 9pm, we'll integrate:

∫T(t)dt=∫56+19sin(πt/12)dt = [56t-19*12/πcos(πt/12)]_t=0^t=12

=[672-228/π*cos(π)]-[0-228/πcos(0)] ≈ 817.14

Now that we have the total value, we will divide by the total time: 9pm-9am=12 hours

average temp = 817.14/12 = 68° F