Sinusoidal temperature model

Text

A city’s daytime temperature is modelled by $T(t)=18+7\sin\!\left(\dfrac{\pi t}{12}\right)$, with $t$ in hours. Find the maximum temperature, minimum temperature and period.

Solution

The sine ranges between $-1$ and $1$, so $7\sin\!\left(\tfrac{\pi t}{12}\right)$ ranges between $-7$ and $7$: $T_{max}=18+7=25\ \text{°C},\qquad T_{min}=18-7=11\ \text{°C}.$ The period of $\sin(\omega t)$ is $\dfrac{2\pi}{\omega}$ with $\omega=\dfrac{\pi}{12}$: $P=\frac{2\pi}{\pi/12}=24\ \text{h}.$ $\boxed{\,T_{max}=25\ \text{°C},\ T_{min}=11\ \text{°C},\ P=24\ \text{h}\,}$