Problem 48 - Calculus with Concepts in Calculus

Question

In a certain industrial area the amount of sulfur dioxide pollutant released into the atmosphere due to burning fossil fuels varies according to the season. Suppose that we wish to model the amount \(A\) of pollutant (in tons) released into the atmosphere at time \(t\) (in wecks) by means of the formula $$ A=1+b \cos \frac{\pi}{26} t $$

Step-by-Step Solution

Answer

The formula models pollutant variation over a year, repeating every 52 weeks, with magnitude determined by \( b \).

Step 1: Understand the Formula

The formula provided is \( A = 1 + b \cos \frac{\pi}{26} t \) where \( A \) represents the amount of pollutant in tons at time \( t \) in weeks, and \( b \) is a constant that affects the amplitude of the cosine function. The term \( \cos \frac{\pi}{26} t \) represents a periodic function with a phase shift determined by \( t \).

Step 2: Analyze the Period of the Cosine Function

The period of a cosine function \( \cos kx \) is given by \( \frac{2\pi}{k} \). In our formula, the argument of the cosine is \( \frac{\pi}{26} t \), meaning \( k = \frac{\pi}{26} \). Hence, the period of the function is \( \frac{2\pi}{\frac{\pi}{26}} = 52 \). This implies the pollution pattern repeats every 52 weeks, which is consistent with annual seasonal changes.

Step 3: Determine the Range of Pollutant Amounts

The range of the cosine function is from -1 to 1. Therefore, \( \cos \frac{\pi}{26} t \) will vary between -1 and 1. Substituting this into the formula \( A = 1 + b \cos \frac{\pi}{26} t \), we find that the range of \( A \) is from \( 1 - b \) to \( 1 + b \).

Step 4: Assess How Constants Affect the Model

The constant \( b \) influences how high and low the pollutant levels go in comparison to the average level of 1 ton. So, larger \( b \) values will cause more significant fluctuations in pollutant amounts, while smaller \( b \) values will result in less variation.

Key Concepts

Periodic Functions

Understanding periodic functions is crucial as they repeat their values in regular intervals. This makes them exceptionally useful in modeling natural and cyclical phenomena, such as the pollutants released into the atmosphere due to seasonal fossil fuel consumption. Periodic functions are characterized by their period, amplitude, and a base function that is often sinusoidal in nature, like sine and cosine functions.

In our example, the formula describes a function that repeats its pattern every 52 weeks. This periodic behavior can be deduced from the cosine function, which naturally repeats after a complete cycle defined by its period. Recognizing the periodic nature of this function helps us understand how pollutant levels might vary consistently over a year.

• The formula: \( A = 1 + b \cos \frac{\pi}{26} t \)
• Period: 52 weeks

Amplitude

Amplitude refers to the height of the wave or oscillation as seen in periodic functions. In the context of our example, it represents how much above or below the average amount of pollution the levels can reach.

The amplitude is determined by the constant \( b \) in the equation. This means larger values of \( b \) will lead to more noticeable highs and lows in the pollution levels, indicating more variability. Conversely, a smaller \( b \) results in pollution levels that stay closer to the average. Understanding amplitude is essential for realizing how varying the constant can lead to different environmental predictions.

• Average level of pollution loss: 1 ton
• Maximum deviation: \( b \) tons
• Range: from \( 1 - b \) to \( 1 + b \) tons

Cosine Function

The cosine function is a fundamental tool in trigonometry, known for its wave-like patterns and its usefulness in plotting periodic phenomena. In the example, the cosine function is used to model the variation in pollutant levels over time.

The function \( \cos \frac{\pi}{26} t \) serves this purpose by allowing the calculation of pollutant levels at any given time, \( t \). This specific function forms peaks and troughs, which correspond to the highest and lowest pollutants released. The cosine function starts its cycle at a peak when \( t = 0 \), reflecting maximum pollutant levels right at the beginning of the cycle.

• Function characteristic: wave
• Repetition pattern: every 52 weeks
• Initial value at \( t = 0 \): peak pollution

Phase Shift

Phase shift is a critical concept when working with trigonometric functions like sine and cosine. It determines the horizontal shift of the wave along the timeline, so it starts or peaks at a different point. In the given example, however, there is no additional term added to create a horizontal shift, which indicates the wave's cycle begins naturally at zero when \( t = 0 \).

In cases where a phase shift is present, it could affect when during the cycle the peak or trough of pollutant levels would occur, potentially aligning these reflective points with specific seasonal events affecting pollution levels.

• Phase shift indicates starting point: naturally at zero
• Effect: would modify timing of peaks and troughs