Problem 70 - Precalculus: Functions and Graphs

Question

The expected low temperature \(T\) (in \(^{\circ} \mathrm{F}\) ) in Fairbanks, Alaska, may be approximated by $$ T=36 \sin \left[\frac{2 \pi}{365}(t-101)\right]+14 $$ where \(t\) is in days, with \(t=0\) corresponding to January 1 For how many days during the year is the low temperature expected to be below \(-4^{\circ} \mathrm{F} ?\)

Step-by-Step Solution

Answer

There are 61 days when the temperature is expected to be below \(-4^{\circ}\)F.

Step 1: Identify the Inequality

To find the days when the temperature is below \(-4^{\circ}\)F, set up the inequality:\[ T < -4 \] Substitute the given formula for \(T\):\[ 36 \sin\left(\frac{2\pi}{365}(t-101)\right) + 14 < -4 \]

Step 2: Simplify the Inequality

Solve the inequality for the sine function:Subtract 14 from both sides:\[ 36 \sin\left(\frac{2\pi}{365}(t-101)\right) < -18 \]Divide both sides by 36:\[ \sin\left(\frac{2\pi}{365}(t-101)\right) < -\frac{1}{2} \]

Step 3: Determine Critical Points

The sine function is \(-\frac{1}{2}\) at angles \(\theta = 7\pi/6\) and \(11\pi/6\) in the interval \([0, 2\pi]\). These critical points correspond to:\[ \theta = \frac{2\pi}{365}(t-101) = \frac{7\pi}{6} \]\[ \theta = \frac{2\pi}{365}(t-101) = \frac{11\pi}{6} \]

Step 4: Solve for Day Values

Solve the critical point equations for \(t\):\[ \frac{2\pi}{365}(t-101) = \frac{7\pi}{6} \]\(t_1 = 365(\frac{7}{12}) + 101 \approx 284.25\).For the second equation:\[ \frac{2\pi}{365}(t-101) = \frac{11\pi}{6} \]\(t_2 = 365(\frac{11}{12}) + 101 \approx 345.92\).Since \(t\) must be an integer, consider \(t_1 = 284 \) and \(t_2 = 345\).

Step 5: Identify the Duration

The days when the sine of the angle is less than \(-\frac{1}{2}\) are between \(t = 284\) and \(t = 345\) for one full cycle.Calculate the duration of these days:\(345 - 284 = 61\).

Key Concepts

Understanding the Sine Function

The sine function, denoted as \( \sin(\theta) \), is an essential component of trigonometry. It helps us describe wave-like phenomena, such as sound waves and, in this context, temperature changes. The sine function is periodic, with values oscillating between -1 and 1. This oscillation is useful for modeling cycles and rhythms occurring in nature.
When applied to this exercise, the sine function is used to model daily temperature changes. The expression \( 36 \sin \left(\frac{2\pi}{365}(t-101)\right) + 14 \) encompasses both the frequency and the amplitude of the temperature variation. Here, \( 36 \) is the amplitude, indicating the range of temperature swings above and below the average. The factor of \( 2\pi/365 \) reflects how the cycle completes each year, capturing the periodic nature of seasonal temperatures, and the shift by \( 101 \) accounts for when the cycle starts, around April 11th. This precise modeling allows us to determine specific conditions, like when the temperature falls below a certain threshold.

Temperature Modeling with Trigonometric Functions

Temperature modeling through trigonometric functions is a method to predict and depict variations in temperature over time. By using the sine model \( T = 36 \sin \left(\frac{2\pi}{365}(t-101)\right) + 14 \), we can anticipate temperature changes seasonally, helping meteorologists and planners make data-driven decisions.
In this scenario, temperatures deviate around a mean temperature of 14°F, peaking up or dropping by up to 36°F from this baseline. The sine function's periodic nature gives insights into the expected fluctuation pattern, predicting when temperatures are likely to exceed or fall below specific points, such as the critical threshold of \(-4°\).
Understanding this model thus facilitates planning for agriculture, energy consumption, and even fashion, as these industries rely heavily on weather predictions.

Periodic Functions and Their Applications

Periodic functions, like the sine function, are defined by their predictable, repeating patterns over intervals. Specifically, the function repeats its values in regular cycles, making them invaluable in modeling cyclical phenomena like seasons, tides, and oscillations.
The exercise illustrates that periodic functions can successfully predict weather patterns. With the sine function's cycle repeating once every 365 days, it mirrors the Earth's orbit around the sun, providing a natural calendar that aligns with the seasonal shifts. This periodicity explains why temperatures return to patterns observed the previous year.
Using periods of these functions allows us to identify specific intervals when particular conditions, such as extremely low or high temperatures, occur. As shown, for days where the sine function is less than \(-1/2\), temperature dips below \(-4°F\), occurring from day 284 to day 345, highlighting a specific window of cold conditions. This knowledge is vital for various sectors, making periodic functions crucial tools in predictive modeling.