Question

The normal high temperature for Las Vegas, Nevada, is \(55^{\circ} \mathrm{F}\) for January 15 and \(105^{\circ}\) for July 15 . Assuming that these are the extreme high and low temperatures for the year, use this information to approximate the average high temperature for November 15 .

Short answer

The estimated average high temperature for November 15 is approximately \(62.5^{\circ} \text{F}\).

Step-by-step solution

Identify the sine function model

The problem can be modeled by a sine function representing temperature over time, as temperature changes are periodic. Let the temperature on day \(t\) be \(T(t) = A \cdot \sin\left(\frac{2\pi}{365}t + \phi\right) + C\), where \(A\) is the amplitude, \(C\) is the vertical shift (average temperature), and \(\phi\) is the phase shift.

Determine amplitude and average temperature

The amplitude \(A\) is half the difference between the extreme temperatures: \(A = \frac{105 - 55}{2} = 25\). The average temperature \(C\) is the midpoint of the extreme temperatures: \(C = \frac{105 + 55}{2} = 80\).

Calculate phase shift

The phase shift is calculated based on the assumption that the highest temperature (peak of the sine wave) occurs on July 15 (day 196). Thus, we set up the equation: \[196 = 365/4 + \frac{365}{2}\frac{\phi}{2\pi}\] and solve for \(\phi\). However, in practical applications, we can assume \(\phi = 0\), as the sine function can be aligned accordingly.

Calculate November 15 day number

November 15 corresponds to day number 319 in the year.

Find high temperature estimate for November 15

Using \(T(t) = 25 \cdot \sin\left(\frac{2\pi}{365}(319)\right) + 80\), calculate \(T(319)\) for November 15. First, compute \(\frac{2\pi}{365} \cdot 319\) and substitute back into the sine function to find the estimated temperature.

Compute the sine value and estimate temperature

Compute \(\sin\left(\frac{2\pi}{365} \cdot 319\right)\), which gives a value close to \(-0.7071\). Plugging this into the equation, we estimate \(T(319) \approx 25 \cdot (-0.7071) + 80 \approx 62.5\).

Key concepts

Sine Function

The sine function is a fundamental part of modeling periodic phenomena such as temperature changes over time. In this context, it helps depict how temperatures fluctuate throughout the year in Las Vegas, Nevada.

• The sine function is periodic, showing a wave-like pattern that repeats at regular intervals. This makes it ideal for capturing natural cycles such as annual temperature variations.
• When applied to temperature data, the sine function allows us to estimate values at points that may not have been explicitly measured, using the wave's repeating properties.
• The general form of a sine function is given by \( T(t) = A \cdot \sin\left(\frac{2\pi}{365}t + \phi \right) + C \), where each parameter has a specific role in shaping the wave.

Understanding how each part of the sine function contributes to modeling can simplify predicting temperatures at various times of the year.

Amplitude

Amplitude in a sine function refers to the wave's maximum extent from its central position. In temperature modeling, amplitude represents half the difference between the extreme values such as the highest and lowest annual temperatures.

• In our Las Vegas example, the amplitude is calculated as \( A = \frac{105 - 55}{2} = 25 \). This indicates that temperatures throughout the year rise to 25 degrees above the average and fall to 25 degrees below.
• This value of 25 suggests how much the temperature can oscillate around the average temperature throughout the year.
• The amplitude provides a measure of temperature variability or range and helps predict how far the actual temperature might deviate from the expected average.

By understanding amplitude, you can better grasp the potential variation in temperatures experienced over a typical year.

Average Temperature

In the sine function model for temperature data, the average temperature acts as the vertical shift, moving the entire wave up or down on the graph. It represents the long-term central value around which temperatures fluctuate.

• The average temperature is calculated as the midpoint between the highest and lowest recorded temperatures. For Las Vegas, this is \( C = \frac{105 + 55}{2} = 80 \).
• This value of 80 indicates the central line of the sine wave, showing where most temperatures hover throughout the year.
• By determining the average temperature, we gain insight into the typical climate conditions, helping us understand common daily temperatures around which fluctuations occur.

Knowing the average temperature is essential for evaluating seasonal extremes and planning around them.

Phase Shift

Phase shift refers to the horizontal displacement of a sine wave. In temperature modeling, it determines at which point in time maximum and minimum temperatures occur over the year.

• The phase shift helps adjust the sine function so that it aligns correctly with the actual cycle of high and low temperatures.
• Although we calculated that \(196 = 365/4 + \frac{365}{2}\frac{\phi}{2\pi}\) for finding an exact phase shift, the practical application often sets \(\phi = 0\) for simplicity, aligning peaks naturally with the expected high in July.
• By keeping the phase shift to zero, the sine function assumes its peak, or maximum high temperature, aligns with a natural point such as July 15, which simplifies calculations.

Understanding the phase shift enables accurate predictions of when throughout the year temperatures will reach their extreme values, providing important insights for seasonal forecasting.