+1904 The Hubble Space Telescope recently took a picture of the currently nearest star to our sun, Proxima Centauri, which is at a distance of\(4.243\) light years (\(1\) light year = \(9,500,000,000,000km\)). The orbits of several other stars near the sun are also known, and during the next\(80,000\) they will replace Proxima Centauri as the nearest star to our sun.
For this problem we will study the two stars Ross 128 and Gliese 445.
The quadratic equations that approximate the distance to each star from our sun in light years are given by:
Gliese 445 : \(D = 0.0104{T}^{2} - 0.942T + 25.382\)
Ross 128 : \(D = 0.0007{T}^{2} - 0.1197T + 11.003\)
where \(T\) is the number of years from the current year in multiples of \(1000\) years.
1) What are the distance ranges for each star over the time interval from \(10,000\) to \(70,000\)in the future?
2) For what values of \(T\), in years, will the distances be identical?
3) What will be the distances to the stars when their distances are identical?
4) When will the two stars be exactly \(3.00\) light years apart sometime over the time range from \(30,000\) to \(80,000\) years from now?
