Problem 15 - Applied Calculus

Question

Delta Cephei is one of the most visible stars in the night sky. Its brightness has periods of \(5.4\) days, the average brightness is \(4.0\) and its brightness varies by \(\pm 0.35 .\) Find a formula that models the brightness of Delta Cephei as a function of time, \(t\), with \(t=0\) at peak brightness.

Step-by-Step Solution

Answer

The formula is \(B(t) = 0.35 \cos\left(\frac{2\pi}{5.4}t\right) + 4.0\).

Step 1: Understand the Problem

We are asked to find a function that models the brightness of Delta Cephei. The star's brightness varies sinusoidally with a period of 5.4 days, a maximum variation of ±0.35 from an average brightness of 4.0. We need to express this variation in a sinusoidal function, which could be either in the form of a sine or cosine function.

Step 2: Identify Parameters for the Sinusoidal Function

Since the sinusoidal function has a peak at \(t = 0\), it's ideal to use a cosine function, as cosine starts at its maximum value. The amplitude \(A\) is 0.35 (the maximum deviation from the average), the average brightness is the vertical shift \(D = 4.0\), and the period \(T = 5.4\) days. The standard form for a cosine function is \(f(t) = A \cos(Bt - C) + D\).

Step 3: Determine the Value of B

The period \(T\) is related to \(B\) by the formula \(\frac{2\pi}{B} = T\). Substituting the given period, we have:\[B = \frac{2\pi}{5.4}\]

Step 4: Write the Sinusoidal Function

Using the identified parameters, the function that models the brightness is:\[B(t) = 0.35 \cos\left(\frac{2\pi}{5.4}t\right) + 4.0\]This formula incorporates the amplitude, period, and vertical shift based on the given values.

Key Concepts

Amplitude

When discussing sinusoidal functions, understanding the concept of amplitude is crucial. Amplitude refers to the maximum distance a wave, like our sinusoidal function, deviates from its central axis.

In our problem, the central axis represents the average brightness of Delta Cephei, which is 4.0. The amplitude is given as \pm 0.35, signifying that the star's brightness can vary by 0.35 above or below its average value.

• The amplitude is a measure of how "tall" or "short" the wave appears.
• In the equation format, it is denoted as \(A\).
• Here, the amplitude \(A = 0.35\), meaning the brightness fluctuates between 3.65 and 4.35.

Amplitude affects the "height" of the oscillation in sinusoidal patterns, representing the range of the star's brightness. With amplitude clearly defined, one can understand how much variation from the mean is present in a sinusoidal wave.

Period of a Function

The period of a function is the interval over which the function repeats itself. For sinusoidal functions describing cycles or waves, the period links with how frequently these cycles complete.

In the scenario of Delta Cephei, the period is provided as 5.4 days. This means the brightness pattern repeats every 5.4 days.

• Period \(T\) directly influences the frequency of repetition for the sinusoidal wave, indicating how fast or slow these cycles occur.
• Mathematically, period is found using \(\frac{2\pi}{B}\).
• Here, \(B\) is determined to ensure the brightness function repeats every 5.4 days, resulting in \(B = \frac{2\pi}{5.4}\).

By understanding the period, one can predict when the sinusoidal function returns to any particular stage in its cycle. It’s a vital component in constructing the wave and ensuring its accurate representation over time.

Cosine Function

The cosine function is one of the primary types of sinusoidal functions used widely in mathematics to model periodic phenomenons.

In our given exercise, the star's brightness is best described by a cosine function as its maximum brightness occurs at \(t=0\). That means the cosine function, which traditionally starts at its peak, is most suitable for this model.

• The general form of a cosine function is given by \(f(t) = A \cos(Bt - C) + D\).
• In this equation:
  • \(A\) is the amplitude, reflecting variation from the average.
  • \(B\) relates to the period and determines how fast the cycles occur.
  • \(C\) is the phase shift; here, \(C=0\) since the peak aligns with \(t=0\).
  • \(D\) is the vertical shift, aligning with the central average, here 4.0.
• In our model, the cosine equation for the brightness is \(B(t) = 0.35 \cos\left(\frac{2\pi}{5.4}t\right) + 4.0\), perfectly capturing the required periodic behavior with its peak at the start of the cycle.

The cosine function, with its defined form and behavior, allows us to precisely depict oscillations like that of Delta Cephei’s brightness. Understanding how to use its parameters correctly is key to accurate mathematical modeling.