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].
