To determine the distance to a Sun-like star with a given luminosity and apparent brightness, we'll use the provided Luminosity Distance Formula:
[tex]\[ \text{Apparent Brightness} (AB) = \frac{\text{Luminosity} (L)}{4 \pi r^2} \][/tex]
Given data:
- Luminosity [tex]\( L = 3.8 \times 10^{26} \)[/tex] watts
- Apparent brightness [tex]\( AB = 1.0 \times 10^{-10} \)[/tex] watt/m²
We need to find the distance [tex]\( r \)[/tex]. First, we'll rearrange the formula to solve for [tex]\( r \)[/tex]:
[tex]\[ r = \sqrt{\frac{L}{4 \pi AB}} \][/tex]
Now, let's plug in the known values:
[tex]\[ r = \sqrt{\frac{3.8 \times 10^{26}}{4 \pi \times 1.0 \times 10^{-10}}} \][/tex]
[tex]\[ r = \sqrt{\frac{3.8 \times 10^{26}}{4 \pi \times 10^{-10}}} \][/tex]
Notice that:
[tex]\[ 4 \pi \approx 12.566 \][/tex]
[tex]\[ r = \sqrt{\frac{3.8 \times 10^{26}}{12.566 \times 10^{-10}}} \][/tex]
[tex]\[ r = \sqrt{\frac{3.8 \times 10^{26}}{1.2566 \times 10^{-9}}} \][/tex]
[tex]\[ r = \sqrt{3.0241 \times 10^{35}} \][/tex]
[tex]\[ r \approx 5.499 \times 10^{17} \, \text{meters} \][/tex]
So, the correct distance to the star is approximately [tex]\( 5.499 \times 10^{17} \)[/tex] meters.
Among the given options, the correct one is:
[tex]\[ 5.499 \times 10^{17} \, \text{m} \][/tex]
