Based on years of weather data, the expected low temperature...

Based on years of weather data, the expected low temperature $T$ (in "F) in Fairbanks, Alaska, can be approximated by $$T=36 \sin \left[\frac{2 \pi}{365}(t-101)\right]+14$$ where $t$ is in days and $t=0$ corresponds to January 1 (a) Sketch the graph of $T$ for $0 \leq t \leq 365$ (b) Predict when the coldest day of the year will occur.

Based on years of weather data, the expected low temperature $T$ (in "F) in Fairbanks, Alaska, can be approximated by
$$T=36 \sin \left[\frac{2 \pi}{365}(t-101)\right]+14$$
where $t$ is in days and $t=0$ corresponds to January 1
(a) Sketch the graph of $T$ for $0 \leq t \leq 365$
(b) Predict when the coldest day of the year will occur.

Key Concepts

Sinusoidal Modeling

This concept involves using sine (or cosine) functions to model periodic phenomena that repeat over a fixed interval, such as seasonal temperature variations. In the context of the problem, the temperature over the year is represented as a sine function, which captures the regular oscillation between high and low temperatures as the days progress.

Parameters of Sinusoidal Functions

Key parameters in a sinusoidal function include the amplitude, period, phase shift, and vertical shift. The amplitude represents the maximum deviation from the midline (or average value), the period is the length of one complete cycle, the phase shift determines the horizontal displacement of the graph, and the vertical shift moves the midline up or down. Understanding these parameters is crucial for interpreting and graphing the function correctly.

Graphing Sinusoidal Functions

Graphing a sinusoidal function involves plotting the midline, maximum and minimum points, and marking key points along one full period of the cycle. By analyzing the amplitude, period, and shifts, one can accurately sketch the waveform over the specified interval. This process is important for visualizing the behavior of periodic models and understanding how specific alterations affect the graph.

Determining Extreme Values

In sinusoidal functions, the extreme values (maximum and minimum) occur at specific points within each cycle. The minimum value is reached when the sine function attains its lowest value, typically -1 for a standard sine curve. Translating this condition into an equation allows one to solve for the time variable at which the phenomenon (temperature, in this case) reaches its extreme. This concept is essential for making predictions about the behavior of periodic phenomena, such as forecasting the coldest day of the year.

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