Average rate of change of temperature over a time interval

First evaluate the function at the endpoints. At $t=6$: $T(6) = 70 + 15\sin(\pi\cdot 6/12) = 70 + 15\sin(\pi/2) = 70 + 15(1) = 85$. At $t=18$: $T(18) = 70 + 15\sin(\pi\cdot 18/12) = 70 + 15\sin(3\pi/2) = 70 + 15(-1) = 55$.

The average rate of change over $[6,18]$ is the slope of the secant line connecting $(6, T(6))$ and $(18, T(18))$: $\frac{T(18)-T(6)}{18-6} = \frac{55 - 85}{12} = \frac{-30}{12} = -2.5.$ So the temperature is decreasing on average by $2.5°F$ per hour over this 12-hour period.