The temperature during the day can be modeled by a sinusoid. Answer the following question given that the low temperature of 52 degrees occurs at 6 AM and the high temperature for the day is 84 degrees. Assuming $t$ is the number of hours since midnight, find an equation for the temperature, $T$, in terms of $t$.

Step 1 ：The temperature can be modeled by a sinusoidal function of the form $T(t)=A\sin (B(t-C))+D$, where $A$ is the amplitude, $B$ is the frequency, $C$ is the phase shift, and $D$ is the vertical shift.

Step 2 ：The amplitude $A$ is half the difference between the high and low temperatures, which is $A=\frac{84-52}{2}=16.0$.

Step 3 ：The vertical shift $D$ is the average of the high and low temperatures, which is $D=\frac{84+52}{2}=68.0$.

Step 4 ：The phase shift $C$ is the time at which the minimum temperature occurs. Since the minimum temperature occurs at 6 AM, $C=6$.

Step 5 ：The period of the sinusoidal function is 24 hours because the temperature pattern repeats every 24 hours. Therefore, the frequency $B$ is the reciprocal of the period, which is $B=\frac{2\pi }{24}=\frac{\pi }{12}$.

Step 6 ：Substituting the values of $A$, $B$, $C$, and $D$ into the equation, we get $T(t)=16.0\sin (\frac{\pi }{12}(t-6))+68.0$.

Step 7 ：$\boxed{{\displaystyle T(t)=16.0\sin (\frac{\pi }{12}(t-6))+68.0}}$ is the equation for the temperature, $T$, in terms of $t$.