Question

during a certain time of year, the daily temperature in a certain city, in degrees celsius follows a periodic pattern. the graph below shows the temperature over two days, where time t is measured in hours after 12:00 a.m. (midnight) on the first day. what is the period and what does it represent in this context?

Explanation:

Step1: Identify period - definition

The period of a periodic function is the smallest positive value \(T\) such that \(f(t + T)=f(t)\) for all \(t\) in the domain of \(f\). In the context of a temperature - time graph, it is the time interval after which the temperature pattern repeats.

Step2: Analyze key points

We have points \((2,12)\), \((14,21)\), \((26,12)\), \((38,21)\). Notice that the temperature values repeat. The time difference between \((2,12)\) and \((26,12)\) (or between \((14,21)\) and \((38,21)\)) gives the period.

Step3: Calculate the period

Subtract the \(t\) - values of two corresponding points. Using the points \((2,12)\) and \((26,12)\), we calculate \(T = 26 - 2=24\) hours. In the context of daily temperature, the period of 24 hours represents one full day. The temperature pattern repeats every 24 hours, which is consistent with the daily cycle of day and night, sunrise and sunset, etc.

Answer:

The period is 24 hours, and it represents the time interval of one full day. The temperature pattern repeats every 24 - hour cycle, which is related to the daily environmental changes such as the rotation of the Earth on its axis causing day and night cycles that affect the temperature.