An astronomer observes the velocity of a galaxy at a distance of $100$ megaparsecs ( $\mathrm{Mpc}$ ) to calculate its distance two billion years from now using two different models. In the first model, the distance from the galaxy to Earth increases by $9 \mathrm{Mpc}$ every billion years. In the second model, the distance from the galaxy to Earth increases by $10$ percent every billion years. Two billion years from now, how much further would the galaxy be in the second model compared to the first model in megaparsecs?

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Q. An astronomer observes the velocity of a galaxy at a distance of

100

megaparsecs (

\mathrm{Mpc}

) to calculate its distance two billion years from now using two different models. In the first model, the distance from the galaxy to Earth increases by

9 \mathrm{Mpc}

every billion years. In the second model, the distance from the galaxy to Earth increases by

10

percent every billion years. Two billion years from now, how much further would the galaxy be in the second model compared to the first model in megaparsecs?

Calculate Distance Increase: Calculate the distance of the galaxy from Earth in the first model after two billion years. $\newline$ The first model predicts an increase of $9 \, \text{Mpc}$ every billion years. Therefore, in two billion years, the increase will be $2 \times 9 \, \text{Mpc}$ . $\newline$ Distance increase in the first model = $2 \times 9 \, \text{Mpc} = 18 \, \text{Mpc}$ .
Calculate New Distance: Calculate the new distance of the galaxy from Earth in the first model after two billion years. $\newline$ Initial distance = $100 \, \text{Mpc}$ . $\newline$ Increase in distance = $18 \, \text{Mpc}$ (from Step $1$ ). $\newline$ New distance in the first model = Initial distance + Increase in distance = $100 \, \text{Mpc} + 18 \, \text{Mpc} = 118 \, \text{Mpc}$ .
Calculate Distance Increase: Calculate the distance of the galaxy from Earth in the second model after one billion years. $\newline$ The second model predicts an increase of $10\%$ every billion years. Therefore, after one billion years, the distance will be increased by $10\%$ of the initial distance. $\newline$ Distance increase after one billion years in the second model = $10\%$ of $100\,\text{Mpc}$ = $0.10 \times 100\,\text{Mpc}$ = $10\,\text{Mpc}$ . $\newline$ New distance after one billion years = Initial distance + Distance increase = $100\,\text{Mpc} + 10\,\text{Mpc}$ = $110\,\text{Mpc}$ .
Calculate New Distance: Calculate the distance of the galaxy from Earth in the second model after the second billion years. $\newline$ Now, we need to increase the new distance after one billion years by another $10$ percent. $\newline$ Distance increase after the second billion years in the second model = $10$ % of $110 \, \text{Mpc} = 0.10 \times 110 \, \text{Mpc} = 11 \, \text{Mpc}$ . $\newline$ New distance after two billion years = Distance after one billion years + Distance increase after the second billion years = $110 \, \text{Mpc} + 11 \, \text{Mpc} = 121 \, \text{Mpc}$ .
Calculate Difference in Distances: Calculate the difference in distances between the two models after two billion years. $\newline$ Distance in the second model after two billion years = $121 \, \text{Mpc}$ (from Step $4$ ). $\newline$ Distance in the first model after two billion years = $118 \, \text{Mpc}$ (from Step $2$ ). $\newline$ Difference in distances = Distance in the second model - Distance in the first model = $121 \, \text{Mpc} - 118 \, \text{Mpc} = 3 \, \text{Mpc}$ .