The temperature during a day can be modeled by a sinusoid. Answer the following question given that the low temperature of 39 degrees occurs at 3 AM and the high temperature for the day is 75 degrees. Find the temperature, to the nearest degree, at 5 PM.

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

Step 2 ：In this case, the amplitude A is half the difference between the high and low temperatures, which is $A=\frac{75-39}{2}=18.0$.

Step 3 ：The frequency B is $\frac{2\pi }{24}$ because the period is 24 hours, which is approximately $B=0.2617993877991494$.

Step 4 ：The phase shift C is the time when the minimum temperature occurs, which is $C=3$.

Step 5 ：The vertical shift D is the average of the high and low temperatures, which is $D=\frac{75+39}{2}=57.0$.

Step 6 ：We can substitute these values into the sinusoidal function to get the temperature at any given time. To find the temperature at 5 PM, we substitute $x=17$ (because 5 PM is 17 hours from 0 AM) into the function.

Step 7 ：Substituting these values into the function, we get $y=18.0\sin (0.2617993877991494(17-3))+57.0$, which simplifies to $y=48$.

Step 8 ：Final Answer: The temperature at 5 PM is $\boxed{{\displaystyle 48}}$ degrees.