Certainly! Let's systematically convert each distance to a common unit, in this case, kilometers, to arrange them in order from least to greatest.
1. Distance to Neptune:
- Already provided in kilometers: [tex]\( 4.7 \times 10^9 \)[/tex] km
2. Distance to Betelgeuse (640 light-years):
- 1 light-year (ly) = 63,000 astronomical units (AU)
- 1 AU = [tex]\( 1.5 \times 10^8 \)[/tex] km
- Convert light-years to astronomical units:
[tex]\( 640 \, \text{ly} \times 63,000 \, \text{AU/ly} = 40,320,000 \, \text{AU} \)[/tex]
- Convert astronomical units to kilometers:
[tex]\( 40,320,000 \, \text{AU} \times 1.5 \times 10^8 \, \text{km/AU} = 6.048 \times 10^{15} \)[/tex] km
- Distance to Betelgeuse: [tex]\( 6.048 \times 10^{15} \)[/tex] km
3. Distance to VY Canis Majoris (3.09 [tex]\(\times 10^8\)[/tex] AU):
- 1 AU = [tex]\( 1.5 \times 10^8 \)[/tex] km
- Convert astronomical units to kilometers:
[tex]\( 3.09 \times 10^8 \, \text{AU} \times 1.5 \times 10^8 \, \text{km/AU} = 4.635 \times 10^{16} \)[/tex] km
- Distance to VY Canis Majoris: [tex]\( 4.635 \times 10^{16} \)[/tex] km
4. Distance to the Large Magellanic Cloud (49,976 parsecs):
- 1 parsec (pc) = 3.262 light-years (ly)
- 1 light-year (ly) = 63,000 astronomical units (AU)
- 1 AU = [tex]\( 1.5 \times 10^8 \)[/tex] km
- Convert parsecs to light-years:
[tex]\( 49,976 \, \text{pc} \times 3.262 \, \text{ly/pc} = 162,949.512 \, \text{ly} \)[/tex]
- Convert light-years to astronomical units:
[tex]\( 162,949.512 \, \text{ly} \times 63,000 \, \text{AU/ly} = 10,265,818,056 \, \text{AU} \)[/tex]
- Convert astronomical units to kilometers:
[tex]\( 10,265,818,056 \, \text{AU} \times 1.5 \times 10^8 \, \text{km/AU} = 1.5405551784 \times 10^{18} \)[/tex] km
- Distance to the Large Magellanic Cloud: [tex]\( 1.5405551784 \times 10^{18} \)[/tex] km
Now, let's arrange these distances from least to greatest:
1. Distance to Neptune: [tex]\( 4.7 \times 10^9 \)[/tex] km
2. Distance to Betelgeuse: [tex]\( 6.048 \times 10^{15} \)[/tex] km
3. Distance to VY Canis Majoris: [tex]\( 4.635 \times 10^{16} \)[/tex] km
4. Distance to the Large Magellanic Cloud: [tex]\( 1.5405551784 \times 10^{18} \)[/tex] km
