Temperature A mathematical model for the temperature in...

Temperature $\quad$ A mathematical model for the temperature in Fairbanks is $$T(t)=37 \sin \left[\frac{2 \pi}{365}(t-101)\right]+25$$ where $T(t)$ is the temperature (in degrees Fahrenheit) on day $t$ with $t=0$ corresponding to January 1 and $t=364$ corresponding to December $31 .$ Use a calculator to estimate the tem- perature for $(a)-(d) .$ Source: The Mathematics Teacher. (a) March 16$($ Day 74$)$ (b) May 2 (Day 121$)$ (c) Day 250 (d) Day 325 (e) Find maximum and minimum values of $T .$ (f) Find the period, T.

Temperature $\quad$ A mathematical model for the temperature in Fairbanks is
$$T(t)=37 \sin \left[\frac{2 \pi}{365}(t-101)\right]+25$$
where $T(t)$ is the temperature (in degrees Fahrenheit) on day $t$ with $t=0$ corresponding to January 1 and $t=364$ corresponding to December $31 .$ Use a calculator to estimate the tem- perature for $(a)-(d) .$ Source: The Mathematics Teacher.
(a) March 16$($ Day 74$)$ (b) May 2 (Day 121$)$ (c) Day 250 (d) Day 325 (e) Find maximum and minimum values of $T .$ (f) Find the period, T.

Key Concepts

Vertical Shift

Vertical shift moves the entire sine function up or down along the vertical axis. This shift adjusts the midline of the function, which represents the average level around which the oscillations occur, tailoring the model to a specific baseline measurement.

Temperature Modeling

Temperature modeling involves using trigonometric functions like the sine wave to represent seasonal temperature variations. By adjusting parameters such as amplitude, period, phase shift, and vertical shift, the model can accurately reflect real-world temperature fluctuations over time.

Phase Shift

Phase shift refers to the horizontal displacement of the sine function along the time axis. It indicates how much the function's cycle is shifted from a standard position, ensuring the model aligns with the correct timing of the observed phenomenon.

Sine Function

A sine function is a mathematical function that produces a smooth, periodic oscillation. It is used to model phenomena that have repetitive cycles, such as seasonal temperature variations, sound waves, and tides.

Amplitude

Amplitude represents the maximum deviation of the function from its midline. In the context of a sine function, it determines the extent of the oscillation above and below the average value, indicating how high or low the function's values can reach.

Period

The period of a sine function is the interval length required for the function to complete one full cycle. It is computed based on the frequency component of the function and is crucial for understanding the cycle length of the modeled phenomenon.

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