Daylight function for 40 °N Verify that the function​​ $D(t)=2.8\sin(\frac{2\pi}{365}(t-81))+12$ has the following properties, where t is measured in days and D is the number of hours between sunrise and sunset. It has a period of 365 days.

Verified step by step guidance

1. Identify the general form of a sinusoidal function, which is typically given by A \sin(B(t - C)) + D, where A is the amplitude, B affects the period, C is the phase shift, and D is the vertical shift.
2. In the given function D(t) = 2.8 \sin\left(\frac{2\pi}{365}(t-81)\right) + 12, compare it to the general form to identify the values of A, B, C, and D. Here, A = 2.8, B = \frac{2\pi}{365}, C = 81, and D = 12.
3. The period of a sinusoidal function is determined by the coefficient B in front of t. The formula for the period is \frac{2\pi}{B}.
4. Substitute B = \frac{2\pi}{365} into the period formula: \text{Period} = \frac{2\pi}{\frac{2\pi}{365}}.
5. Simplify the expression for the period to verify that it equals 365 days, confirming that the function has the desired period.