The farthest objects (known as quasers) in our universe are so distant that light emitted by them takes billion of years to reach the earth. What is the distance in km of a quaser from which light takes 3.0 billion years to reach us ?

Correct Option: 2.84 x 10²² km

To find the distance of the quasar in kilometers, we can use the relation between distance, speed, and time:

$\text{Distance}=\text{Speed of light}\times \text{Time}$

Let's first write down the given values and convert them into compatible units (seconds for time and kilometers per second for the speed of light):

1. Speed of light ($c$):
The speed of light in a vacuum is approximately:
$c=3.0\times {10}^8\text{ m/s}$
Converting this speed to kilometers per second ($\text{km/s}$):
$c=3.0\times {10}^5\text{ km/s}$

2. Time taken ($t$):
The light takes 3.0 billion years to reach the Earth.
$t=3.0\text{ billion years}=3.0\times {10}^9\text{ years}$
Now, we convert years into seconds (assuming 1 year = 365 days):
$t=3.0\times {10}^9\times 365\times 24\times 3600\text{ seconds}$
$t\approx 3.0\times {10}^9\times 3.1536\times {10}^7\text{ seconds}$
$t\approx 9.46\times {10}^{16}\text{ seconds}$

3. Calculating the Distance ($d$):
Substitute the values of speed and time into the distance formula:
$d=c\times t$
$d=(3.0\times {10}^5\text{ km/s})\times (9.46\times {10}^{16}\text{ s})$
$d=28.38\times {10}^{21}\text{ km}$
$d\approx 2.84\times {10}^{22}\text{ km}$

Therefore, the distance of the quasar from Earth is approximately 2.84 x 10²² km.