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A Cepheid star is a type of variable star, which means its brightness is not constant. The relationship between the brightness of a Cepheid star and its period, or length of its pulse, is given by:

[tex]\[ M = -2.78(\log P) - 1.35 \][/tex]

where [tex]\( M \)[/tex] is the absolute magnitude (brightness) of the star, and [tex]\( P \)[/tex] 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. [tex]\(-5.95\)[/tex]
B. [tex]\(-4.60\)[/tex]
C. [tex]\(3.25\)[/tex]
D. [tex]\(4.60\)[/tex]

Sagot :

GinnyAnsweridn
To determine the absolute magnitude [tex]\( M \)[/tex] of a Cepheid star with a period of 45 days using the given formula:

[tex]\[ M = -2.78 (\log P) - 1.35 \][/tex]

Follow these steps:

1. Calculate [tex]\(\log P\)[/tex] where [tex]\( P = 45 \)[/tex]:

First, find the base-10 logarithm of 45.
[tex]\[ \log 45 \approx 1.6532 \][/tex]

2. Substitute [tex]\(\log P \)[/tex] into the formula:

Now, substitute the value of [tex]\(\log 45\)[/tex] into the magnitude formula:
[tex]\[ M = -2.78 (1.6532) - 1.35 \][/tex]

3. Simplify the expression:

Perform the multiplication:
[tex]\[ -2.78 \times 1.6532 \approx -4.5918 \][/tex]

Then, add the constant term:
[tex]\[ M = -4.5918 - 1.35 = -5.9418 \][/tex]

4. Round to the nearest hundredth:

Finally, round the result to the nearest hundredth to get the absolute magnitude:
[tex]\[ M \approx -5.95 \][/tex]

So, the absolute magnitude of a Cepheid star with a period of 45 days is approximately [tex]\(-5.95\)[/tex].

Therefore, the correct answer is [tex]\(\boxed{-5.95}\)[/tex].
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