Answered

The relationship between the brightness of a Cepheid star and its period, or length of its pulse, is given by $M=-2.78(\log P)-1.35$, where $M$ is the absolute magnitude, or brightness, of the star, and $P$ is the number of days required for the star to complete one cycle. What is the absolute magnitude of a star that has a period of 45 days? Use a calculator. Round your answer to the nearest hundredth.


A-5.95
B-4.60
C3.25
D4.60

Verified Answer :

  • Start with the given formula: \(M = -2.78(\log_{10} P) - 1.35\).
  • Substitute \(P = 45\) into the equation to get \(M = -2.78(\log_{10} 45) - 1.35\).
  • Calculate the numerical value to be approximately \(-5.94592\).
  • Round the result to the nearest hundredth, giving \(\boxed{-5.95}\).

Explanation

  1. Understanding the Problem and Data Analysis

The problem provides the relationship between the absolute magnitude \(M\) of a Cepheid star and its pulsation period \(P\) (in days) using the formula \(M = -2.78(\log_{10} P) - 1.35\). We are given a specific period of \(P = 45\) days and are tasked with finding the corresponding absolute magnitude \(M\), rounded to the nearest hundredth. This formula is a logarithmic model, which is common in astrophysics to relate observable quantities like periodicity to intrinsic properties like brightness.

  1. Calculating the Absolute Magnitude

To calculate the value of \(M\), we substitute \(P = 45\) into the given formula:
\[M = -2.78(\log_{10} 45) - 1.35\]
First, we calculate the base-10 logarithm of 45. Using logarithmic tables or a calculator, we find that:
\[\log_{10} 45 \approx 1.65321\]
Next, we perform the multiplication:
\[-2.78 \times 1.65321 \approx -4.59592\]
Finally, we subtract 1.35 from this result:
\[M \approx -4.59592 - 1.35 = -5.94592\]

  1. Rounding and Selecting the Correct Answer

We have calculated the absolute magnitude \(M\) to be approximately \(-5.94592\). The question asks for the answer rounded to the nearest hundredth. Looking at the third decimal place, which is 5, we round the second decimal place (the 4) up to a 5. Thus, the value becomes \(-5.95\). Comparing this result with the given choices, we find it matches the first option.

Examples

Logarithmic scales are used throughout science and engineering to describe phenomena that span many orders of magnitude. For example, the magnitude scale in astronomy (as seen here) is logarithmic, as is the Richter scale for earthquakes, the decibel scale for sound intensity, and the pH scale for acidity. By using logarithms, scientists can represent extremely large or small values as manageable, small numbers, allowing for easier analysis of trends in data like the brightness of distant stars compared to their cycle lengths.

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