Alright, let's solve the problem using the given formula for apparent brightness.
Given:
- Apparent Brightness [tex]\((A B) = 3.2 \times 10^{-10} \text{ watt/m}^2\)[/tex]
- Distance [tex]\(r = 3 \times 10^{17} \text{ meters}\)[/tex]
The formula for apparent brightness is:
[tex]\[
A B = \frac{L}{4 \pi r^2}
\][/tex]
Where:
- [tex]\(A B\)[/tex] is the apparent brightness
- [tex]\(L\)[/tex] is the luminosity
- [tex]\(r\)[/tex] is the distance
We need to find the luminosity [tex]\(L\)[/tex]. We can rearrange the formula to solve for [tex]\(L\)[/tex]:
[tex]\[
L = A B \times 4 \pi r^2
\][/tex]
Now, let’s plug in the given values:
[tex]\[
L = (3.2 \times 10^{-10} \text{ watt/m}^2) \times 4 \pi (3 \times 10^{17} \text{ meters})^2
\][/tex]
Let's break it down step-by-step:
1. Calculate the distance squared:
[tex]\[
r^2 = (3 \times 10^{17} \text{ meters})^2 = 9 \times 10^{34} \text{ meters}^2
\][/tex]
2. Multiply by [tex]\(4 \pi\)[/tex]:
[tex]\[
4 \pi r^2 = 4 \pi \times 9 \times 10^{34} \text{ meters}^2
\][/tex]
Using the approximation [tex]\(\pi \approx 3.141592\)[/tex]:
[tex]\[
4 \pi \approx 12.5664
\][/tex]
[tex]\[
4 \pi \times 9 \times 10^{34} = 12.5664 \times 9 \times 10^{34} \approx 113.0976 \times 10^{34}
\][/tex]
[tex]\[
113.0976 \times 10^{34} = 1.130976 \times 10^{36}
\][/tex]
3. Multiply by the apparent brightness:
[tex]\[
L = 3.2 \times 10^{-10} \text{ watt/m}^2 \times 1.130976 \times 10^{36} \text{ meters}^2 = 3.6191248 \times 10^{26} \text{ watts}
\][/tex]
After rounding off to match the options provided, we get:
[tex]\[
L \approx 3.619 \times 10^{26} \text{ watts}
\][/tex]
Thus, the luminosity of the star is:
[tex]\[
\boxed{3.619 \times 10^{26} \text{ watts}}
\][/tex]
