Problem 46 - Contemporary Precalculus

Question

The brightness of the binary star Beta Lyrae (as seen from the earth) varies. Its visual magnitude \(M(t)\) after \(t\) days is approximately $$M(t)=.55 \cos (.97 t)+3.85$$ The visual magnitude scale is reversed from what you would expect: The lower the number, the brighter the star. With this in mind, answer the following questions. (a) Graph the function \(M\) when \(0 \leq t \leq 21\) (b) What is the visual magnitude when the star is brightest? When it is dimmest? (c) What is the period of the magnitude (the interval between its brightest times)?

Step-by-Step Solution

Answer

Answer: The period of the magnitude of the binary star Beta Lyrae is approximately 6.47 days.

Step 1: Recognize the given function

The given function is \(M(t) = 0.55 \cos(0.97t) + 3.85\). This is a cosine function with amplitude 0.55, period \(\frac{2\pi}{0.97}\), and vertical shift 3.85.

Step 2: Set up the graph

Using graphing software such as Desmos or a graphing calculator, input the function \(M(t) = 0.55 \cos(0.97t) + 3.85\) for the range \(0 \leq t \leq 21\). Then, plot the graph to visualize the function. #b) Visual magnitude when the star is brightest and dimmest#

Step 1: Identify the maximum and minimum visual magnitude

To identify the maximum and minimum visual magnitude for the given function, consider the amplitude and the vertical shift of the function. The amplitude is 0.55 and the vertical shift is 3.85. The maximum visual magnitude occurs when the cosine function is 1, and the minimum occurs when the cosine function is -1.

Step 2: Calculate the maximum and minimum magnitude

To calculate the maximum and minimum visual magnitude, we plug the maximum and minimum cosine function values into the equation. Maximum: \(M(t) = 0.55(1) + 3.85 = 4.4\) Minimum: \(M(t) = 0.55(-1) + 3.85 = 3.3\)

Step 3: Express the brightest and dimmest visual magnitudes

To express the brightest and dimmest visual magnitudes, remember that for visual magnitude, lower numbers indicate a brighter star. Brightest (minimum visual magnitude): \(M(t) = 3.3\) Dimmest (maximum visual magnitude): \(M(t) = 4.4\) #c) Period of magnitude#

Step 1: Find the period of the cosine function

Recall the standard cosine function has a period of \(2\pi\). Since our function is \(M(t) = 0.55 \cos(0.97t) + 3.85\), the cosine function has a coefficient 0.97 for the variable \(t\). The period is given by the formula \(\frac{2\pi}{k}\) where \(k\) is the coefficient of \(t\).

Step 2: Calculate the period of the magnitude

To find the period of the magnitude of the star, use the formula \(\frac{2\pi}{0.97}\). Period of magnitude: \(\frac{2\pi}{0.97} \approx 6.47\) days The period of the magnitude of the binary star Beta Lyrae is approximately 6.47 days.

Key Concepts

Amplitude

When dealing with trigonometric functions such as the cosine function, amplitude is a critical concept. Amplitude represents the distance from the middle of the function's range to its highest or lowest point. This determines how "tall" or "short" the waves of the function appear on a graph. In the context of the brightness of a binary star like Beta Lyrae, the amplitude tells us how much the apparent visual magnitude varies from its average. For our function, the amplitude is given as 0.55.

Understanding this, we know that the visual magnitude of the star changes by 0.55 units above and below the average value of magnitude, which is affected by external factors, such as how the two stars obscures each other. Hence, whenever calculating or interpreting trigonometric functions involving celestial brightness, keenly observe the amplitude to know the extent of variation in brightness.

Period of Trigonometric Function

The period of a trigonometric function is the interval over which the function completes one full cycle. For a standard cosine function, the period is generally \(2\pi\). However, when the cosine function is modified by a coefficient affecting the variable inside its argument, the period will change.

In the case of our function, where the argument is \(0.97t\), the coefficient of \(t\) is \(0.97\). The formula to find the new period is \( \frac{2\pi}{k} \), where \(k\) is the coefficient of \(t\). Applying this to our problem gives us a period of approximately 6.47 days. This tells us that every 6.47 days, the visual magnitude of Beta Lyrae will complete a cycle of going from its brightest to its dimmest state and back again.

Hence, understanding the period is essential for predicting repetitive changes over time, particularly useful when studying periodic astronomical phenomena.

Cosine Function

The cosine function is one of the fundamental trigonometric functions, and it plays a significant role in modeling periodic phenomena because of its wave-like characteristics. In its standard form, the cosine function is written as \(\cos(x)\), featuring a smooth, oscillating pattern that repeats every \(2\pi\) units.

For the function related to Beta Lyrae, the cosine component \(\cos(0.97t)\) was used to describe the star's changing brightness over time. Here, the argument of the cosine modifies the function's standard period and the amplitude reflects how much the brightness changes. With a cosine function, the highest value is reached when it is at 1, and the lowest when it is at -1.

This behavior is particularly useful in astronomical applications since it provides a simplified yet powerful means of predicting periodic changes in brightness as celestial objects move and rotate in space. Understanding how the cosine function operates in such contexts is key to grasping more complex aspects of trigonometry and its applications.