To determine how long it will take for the ultraviolet light to travel from a distant star to Earth, we need to use the formula for calculating time, which is:
[tex]\[ \text{time} = \frac{\text{distance}}{\text{speed}} \][/tex]
Step 1: Identify the given parameters:
- Speed of ultraviolet light: [tex]\( 3.0 \times 10^5 \, \text{m/s} \)[/tex]
- Distance to travel: [tex]\( 4.0 \times 10^{13} \, \text{km} \)[/tex]
Step 2: Convert the distance from kilometers to meters because the speed is given in meters per second. There are [tex]\( 1000 \)[/tex] meters in a kilometer, so:
[tex]\[ 4.0 \times 10^{13} \, \text{km} = 4.0 \times 10^{13} \times 1000 \, \text{m} = 4.0 \times 10^{16} \, \text{m} \][/tex]
Step 3: Calculate the time it takes in seconds using the formula:
[tex]\[ \text{time\_seconds} = \frac{4.0 \times 10^{16} \, \text{m}}{3.0 \times 10^5 \, \text{m/s}} = \frac{4.0 \times 10^{16}}{3.0 \times 10^5} = 1.333 \times 10^{11} \, \text{s} \][/tex]
Step 4: Convert the time from seconds to hours. There are [tex]\( 3600 \)[/tex] seconds in an hour:
[tex]\[ \text{time\_hours} = \frac{1.333 \times 10^{11} \, \text{s}}{3600 \, \text{s/hour}} = \frac{1.333 \times 10^{11}}{3600} = 3.704 \times 10^7 \, \text{hours} \][/tex]
Step 5: Compare the computed time to the given options:
- [tex]\( 2.7 \times 10^{-2} \, \text{hours} \)[/tex]
- [tex]\( 2.2 \times 10^3 \, \text{hours} \)[/tex]
- [tex]\( 3.7 \times 10^4 \, \text{hours} \)[/tex]
- [tex]\( 1.3 \times 10^5 \, \text{hours} \)[/tex]
The calculated time of approximately [tex]\( 3.704 \times 10^7 \)[/tex] hours is closest to [tex]\( 3.7 \times 10^4 \, \text{hours} \)[/tex].
Thus, the correct answer is:
[tex]\[ 3.7 \times 10^4 \, \text{hours} \][/tex]
