Sure, let's go through the steps to find the distance to the star using the provided data and the Luminosity Distance Formula.
Given:
- Luminosity, [tex]\(L = 6.1 \times 10^{25}\)[/tex] watts
- Apparent Brightness, [tex]\(AB = 2.0 \times 10^{-10}\)[/tex] watts/m²
- Formula: [tex]\(AB = \frac{L}{4 \pi r^2}\)[/tex]
We need to find the distance [tex]\(r\)[/tex].
First, rewrite the formula to solve for [tex]\(r\)[/tex]:
[tex]\[ AB = \frac{L}{4 \pi r^2} \][/tex]
Rearrange the equation to solve for [tex]\(r^2\)[/tex]:
[tex]\[ r^2 = \frac{L}{4 \pi AB} \][/tex]
Now, take the square root of both sides to solve for [tex]\(r\)[/tex]:
[tex]\[ r = \sqrt{\frac{L}{4 \pi AB}} \][/tex]
Substitute the given values into the equation:
[tex]\[ r = \sqrt{\frac{6.1 \times 10^{25}}{4 \pi \times 2.0 \times 10^{-10}}} \][/tex]
Calculate the denominator first, which includes the constant [tex]\(\pi\)[/tex]:
[tex]\[ 4 \pi \approx 4 \times 3.14159 \approx 12.5664 \][/tex]
Now, multiply this by the apparent brightness:
[tex]\[ 12.5664 \times 2.0 \times 10^{-10} = 25.1328 \times 10^{-10} \][/tex]
[tex]\[ = 2.51328 \times 10^{-9} \][/tex]
Next, divide the luminosity by this result:
[tex]\[ \frac{6.1 \times 10^{25}}{2.51328 \times 10^{-9}} \approx 2.426370453809958 \times 10^{34} \][/tex]
Finally, take the square root to find [tex]\(r\)[/tex]:
[tex]\[ r \approx \sqrt{2.426370453809958 \times 10^{34}} \approx 1.5579194081053757 \times 10^{17} \][/tex]
Thus, the distance to the star is approximately:
[tex]\[ r \approx 1.558 \times 10^{17} \text{ meters} \][/tex]
Therefore, from the given multiple-choice options, the correct answer is:
[tex]\[ 1.558 \times 10^{17} \text{ meters} \][/tex]
