To arrange the distances from Earth to various celestial objects in order from least to greatest, we need to convert all given distances to a common unit, such as kilometers. Here are the given distances and their units:
1. Distance to Neptune (km): [tex]\( 4.7 \times 10^9 \)[/tex] km
2. Distance to Betelgeuse (ly): 640 ly
3. Distance to VY Canis Majoris (AU): [tex]\( 3.09 \times 10^8 \)[/tex] AU
4. Distance to the Large Magellanic Cloud (pc): 49976 pc
Let's convert each distance to kilometers step-by-step:
### 1. Neptune
The distance to Neptune is already given in kilometers:
[tex]\[ 4.7 \times 10^9 \text{ km} \][/tex]
### 2. Betelgeuse
The distance to Betelgeuse is given in light years. We need to convert light years to kilometers.
- 1 light year = 63,000 AU
- 1 AU = [tex]\( 1.5 \times 10^8 \)[/tex] km
First, convert light years to AU:
[tex]\[ 640 \text{ ly} = 640 \times 63,000 \text{ AU} \][/tex]
[tex]\[ 640 \times 63,000 = 40,320,000 \text{ AU} \][/tex]
Now, convert AU to kilometers:
[tex]\[ 40,320,000 \text{ AU} = 40,320,000 \times 1.5 \times 10^8 \text{ km} \][/tex]
[tex]\[ 40,320,000 \times 1.5 \times 10^8 = 6.048 \times 10^{15} \text{ km} \][/tex]
### 3. VY Canis Majoris
The distance to VY Canis Majoris is given in Astronomical Units (AU). Convert AU to kilometers:
[tex]\[ 3.09 \times 10^8 \text{ AU} = 3.09 \times 10^8 \times 1.5 \times 10^8 \text{ km} \][/tex]
[tex]\[ = 4.635 \times 10^{16} \text{ km} \][/tex]
### 4. Large Magellanic Cloud
The distance to the Large Magellanic Cloud is given in parsecs (pc). Convert parsecs to kilometers.
- 1 parsec = 3.262 light years
- 1 light year = 63,000 AU
- 1 AU = [tex]\( 1.5 \times 10^8 \)[/tex] km
First, convert parsecs to light years:
[tex]\[ 49976 \text{ pc} = 49976 \times 3.262 \text{ ly} \][/tex]
[tex]\[ 49976 \times 3.262 = 162948.912 \text{ ly} \][/tex]
Next, convert light years to AU:
[tex]\[ 162948.912 \text{ ly} = 162948.912 \times 63,000 \text{ AU} \][/tex]
[tex]\[ 162948.912 \times 63,000 = 10,266,781,456 \text{ AU} \][/tex]
Finally, convert AU to kilometers:
[tex]\[ 10,266,781,456 \text{ AU} = 10,266,781,456 \times 1.5 \times 10^8 \text{ km} \][/tex]
[tex]\[ = 1.5405551784 \times 10^{18} \text{ km} \][/tex]
### Arranging Distances
Now we list the distances in kilometers from least to greatest:
1. Neptune: [tex]\( 4.7 \times 10^9 \)[/tex] km
2. Betelgeuse: [tex]\( 6.048 \times 10^{15} \)[/tex] km
3. VY Canis Majoris: [tex]\( 4.635 \times 10^{16} \)[/tex] km
4. Large Magellanic Cloud: [tex]\( 1.5405551784 \times 10^{18} \)[/tex] km
The distances in order from least to greatest are:
1. Neptune [tex]\( (4.7 \times 10^9 \text{ km}) \)[/tex]
2. Betelgeuse [tex]\( (6.048 \times 10^{15} \text{ km}) \)[/tex]
3. VY Canis Majoris [tex]\( (4.635 \times 10^{16} \text{ km}) \)[/tex]
4. Large Magellanic Cloud [tex]\( (1.5405551784 \times 10^{18} \text{ km}) \)[/tex]
