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 \(\boxed{48}\) degrees.