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

T(t) = 52 + 15 sin(πt/12)
Find the average temperature Tave during the period from 9 AM to 9 PM. (Round your answer to the nearest whole number.)

To find the average temperature Tave during the period from 9 AM to 9 PM, we need to integrate the temperature function T(t) over the interval from 0 to 12.

The formula for the average of a function over an interval [a, b] is:

Tave = (1 / (b - a)) * ∫[a,b] T(t) dt

In our case, a = 0 and b = 12. Therefore, the formula becomes:

Tave = (1 / (12 - 0)) * ∫[0,12] T(t) dt

Tave = (1 / 12) * ∫[0,12] (52 + 15 sin(πt/12)) dt

Now, let's integrate the function using the power rule and the antiderivative of sin(x):

Tave = (1 / 12) * [52t - 180 cos(πt/12)] evaluated from 0 to 12

Tave = (1 / 12) * [(52 * 12 - 180 cos(12π/12)) - (52 * 0 - 180 cos(0π/12))]

Tave = (1 / 12) * [624 - 180 cos(π)]

Using the approximation for cos(π) = -1, we have:

Tave = (1 / 12) * (624 - 180 * (-1))

Tave = (1 / 12) * (624 + 180)

Tave = (1 / 12) * 804

Tave ≈ 67

Therefore, the average temperature during the period from 9 AM to 9 PM is approximately 67°F.