Let's arrange the distances from Earth to various celestial objects in order from least to greatest. We'll use the provided conversion table to convert all given distances to kilometers (km).
### Given Data:
1. Distance to Betelgeuse: [tex]\( 640 \)[/tex] light years (ly)
2. Distance to W Canis Majoris: [tex]\( 3.09 \times 10^8 \)[/tex] Astronomical Units (AU)
3. Distance to the Large Magellanic Cloud (LMC): [tex]\( 49976 \)[/tex] parsecs (pc)
4. Distance to Neptune: [tex]\( 4.7 \)[/tex] billion km
### Conversion Table:
1. [tex]\( 1 \)[/tex] AU = [tex]\( 1.5 \times 10^8 \)[/tex] km
2. [tex]\( 1 \)[/tex] ly = [tex]\( 63000 \)[/tex] AU
3. [tex]\( 1 \)[/tex] pc = [tex]\( 3.262 \)[/tex] ly
Let's convert each distance to kilometers:
### 1. Distance to Betelgeuse:
[tex]\[
\text{Distance (km)} = 640 \, \text{ly} \times 63000 \, \text{AU/ly} \times 1.5 \times 10^8 \, \text{km/AU}
\][/tex]
[tex]\[
= 640 \times 63000 \times 1.5 \times 10^8 \, \text{km}
\][/tex]
[tex]\[
= 6.048 \times 10^{15} \, \text{km}
\][/tex]
### 2. Distance to W Canis Majoris:
[tex]\[
\text{Distance (km)} = 3.09 \times 10^8 \, \text{AU} \times 1.5 \times 10^8 \, \text{km/AU}
\][/tex]
[tex]\[
= 4.635 \times 10^{16} \, \text{km}
\][/tex]
### 3. Distance to the Large Magellanic Cloud:
[tex]\[
\text{Distance (km)} = 49976 \, \text{pc} \times 3.262 \, \text{ly/pc} \times 63000 \, \text{AU/ly} \times 1.5 \times 10^8 \, \text{km/AU}
\][/tex]
[tex]\[
= 49976 \times 3.262 \times 63000 \times 1.5 \times 10^8 \, \text{km}
\][/tex]
[tex]\[
= 1.5405551784 \times 10^{18} \, \text{km}
\][/tex]
### 4. Distance to Neptune:
It's already given in kilometers:
[tex]\[
\text{Distance (km)} = 4.7 \times 10^9 \, \text{km}
\][/tex]
### Sorting the Distances:
1. Neptune: [tex]\( 4.7 \times 10^9 \)[/tex] km
2. Betelgeuse: [tex]\( 6.048 \times 10^{15} \)[/tex] km
3. W Canis Majoris: [tex]\( 4.635 \times 10^{16} \)[/tex] km
4. Large Magellanic Cloud: [tex]\( 1.5405551784 \times 10^{18} \)[/tex] km
### Final Ordered List:
1. Neptune: [tex]\( 4.7 \times 10^9 \)[/tex] km
2. Betelgeuse: [tex]\( 6.048 \times 10^{15} \)[/tex] km
3. W Canis Majoris: [tex]\( 4.635 \times 10^{16} \)[/tex] km
4. Large Magellanic Cloud: [tex]\( 1.5405551784 \times 10^{18} \)[/tex] km
