Question

over the course of a full year, the daylight in a certain city follows a periodic pattern. the graph below represents the average daylight, in minutes, over the course of twenty - four months, with time t representing the number of months after january 1 of a certain year. what is the amplitude and what does it represent in this context? average daylight (in minutes) time (in months)

Explanation:

Step1: Recall amplitude formula

The amplitude $A$ of a periodic - function $y = A\sin(Bx - C)+D$ (or similar periodic forms) is given by $A=\frac{\text{Max}-\text{Min}}{2}$, where $\text{Max}$ is the maximum value of the function and $\text{Min}$ is the minimum value of the function.

Step2: Identify Max and Min values

From the graph, the maximum value of the average - daylight function is $y_{max}=863$ minutes and the minimum value is $y_{min}=515$ minutes.

Step3: Calculate the amplitude

Using the amplitude formula $A=\frac{y_{max}-y_{min}}{2}$, we substitute $y_{max}=863$ and $y_{min}=515$:
\[A=\frac{863 - 515}{2}=\frac{348}{2}=174\]

Step4: Interpret the amplitude

In the context of average daylight, the amplitude represents half of the difference between the maximum and minimum amounts of average daylight in a year. It measures the deviation of the average - daylight amount from the mid - value of the daylight cycle over a year.

Answer:

The amplitude is 174 minutes. It represents half of the difference between the maximum and minimum amounts of average daylight in a year for the certain city.