Problem 64 - Calculus Single Variable

Question

The lowest monthly normal temperature of Philadelphia is \(31^{\circ} \mathrm{F}\) and occurs in January. The highest monthly normal temperature of Philadelphia is \(77^{\circ} \mathrm{F}\) and occurs in July. Find a model of temperature \(T\) as a function of time that has the form \(T(t)=b+A \sin (\omega t+\phi)\).

Step-by-Step Solution

Answer

The model for Philadelphia's monthly temperature is \( T(t) = 54 + 23 \sin \left( \frac{\pi}{6}t - \frac{2\pi}{3} \right) \).

Step 1: Determine the average temperature

The average temperature, or the vertical shift, of the sine function can be calculated as the mean of the highest and lowest temperatures. Calculate it using the formula: \( b = \frac{77 + 31}{2} \). This gives us \( b = 54 \).

Step 2: Calculate the amplitude of the function

The amplitude \( A \) is half the difference between the maximum and minimum temperatures. Calculate it using the formula: \( A = \frac{77 - 31}{2} \). This yields \( A = 23 \).

Step 3: Determine the period of the function

The period of the temperature variation is a year, which we consider as 12 months. The angular frequency \( \omega \) is determined based on the period: \( 2\pi/\text{period} = \omega = \frac{2\pi}{12} = \frac{\pi}{6} \).

Step 4: Calculate the phase shift

The function should reach its minimum at \( t = 1 \) (January) and its maximum at \( t = 7 \) (July). For a typical sine function, the minimum occurs at \( -\frac{\pi}{2} \). Hence, we solve \( -\frac{\pi}{2} = \omega \cdot 1 + \phi \) for \( \phi \). Using \( \omega = \frac{\pi}{6} \), calculate \( \phi = -\frac{\pi}{2} - \frac{\pi}{6} \). This results in \( \phi = -\frac{2\pi}{3} \).

Step 5: Write the model equation

We now have all the components to write the temperature function: \( T(t) = 54 + 23 \sin \left( \frac{\pi}{6}t - \frac{2\pi}{3} \right) \). This function models the temperature as a sine wave with the components we've calculated.

Key Concepts

Temperature Modeling

Temperature modeling allows us to predict how temperature changes over time using mathematical functions. Specifically, in this scenario, we employ the sine function due to its wave-like properties resembling naturally occurring temperature fluctuations.

• Purpose: Modeling helps in understanding seasonal changes. It enables the prediction of temperatures across different times of the year.
• Variables: In the given model, time typically represents months and temperature is the dependent variable modeled over a year.
• Benefit of using sine: Sine functions describe periodic phenomena, making them fit well for temperature variations that repeat annually.

To construct a successful model, we need parameters like amplitude, phase shift, vertical shift, and angular frequency. These elements shape the sine function to match real-world temperature patterns.
Each component has its specific place in the equation, and together they form a cohesive model that aptly represents the temperature changes over time.

Sine Function

The sine function is fundamental in trigonometry and used widely in temperature modeling due to its repetitive wave pattern.

• Form: A sine function is typically expressed as: \( y = A \sin(\omega t + \phi) + b \).
• Graph characteristics: It oscillates between a maximum and minimum value, which makes it ideal for modeling temperature that generally has high and low points.
• Periodicity: The function repeats its shape at specified intervals, meaning it cycles every fixed number of units (e.g., months in our model).

The sine function used in the temperature model takes into account different parameters: amplitude for the variation range, phase shift for the starting point of the cycle, and a vertical shift representing the average value.
Understanding the sine function's potential to describe periodic motion is essential, as this pattern vividly aligns with natural temperature variations over months and seasons.

Amplitude

Amplitude is a major component of any sine function and plays a crucial role in temperature modeling. It describes the extent of the temperature change over its average.

• Definition: Amplitude is the distance from the midline of the sine function to its maximum or minimum. It essentially measures how far the temperature can deviate from the average.
• Calculation: In the given problem, amplitude was calculated as half the difference between the highest and lowest temperatures, \( A = \frac{77 - 31}{2} = 23 \).
• Impact: A larger amplitude indicates more extreme temperature variations, which might signify a high range for temperature changes throughout the year.

In temperature modeling, carefully determining the amplitude ensures that the predicted model can accurately reflect the real-world range of temperature differences.

Phase Shift

Phase shift refers to the horizontal movement of the sine wave on the time axis. It's essential for aligning the model with a real-world cycle.

• Purpose: It adjusts the starting point of the sine function to match the actual time of temperature extremes. In our case, ensuring that January and July temperatures align with the function's min and max, respectively.
• Finding phase shift: For our model, we need to set the sine's minimum in January, resulting in the calculation: \( -\frac{\pi}{2} = \omega \cdot 1 + \phi \) leading to \( \phi = -\frac{2\pi}{3} \).
• Effects: Incorrect phase shift can lead to misalignment, making the predictions inaccurate, which demonstrates how critical this calculation is.

Understanding phase shifts in trigonometric functions is vital for accurate modeling. It ensures the predicted temperature shifts correspond accurately to real changes through the year.