In this problem, Chapter 28 number 7 in Cutnell Volume 7, I am getting the prope

Question

In this problem, Chapter 28 number 7 in Cutnell Volume 7, I am getting the proper and dilated times mixed up.  Wouldn't the earth observer see the dilated time of the bacterium of the ship, not the proper time?  I thought for proper time you had to observe the two events (bacterium from day 1-day x) at rest with respect to the events?  The observer on earth is seeing the ship taking off, and then the bacterium at some time x.  Wouldn't that be considered dilated time?  This problem solution seems to be the opposite, and I don't understand why?

Explanation / Answer

   as from the given    to = 24.0 hr    so for the earth based sample to grow to 512bacteria it would take    2n = 512    n = 9    t = to / [1 -(v / c)2]         = 24.0 h /[1 - (0.807 c / c)2]         = ........ h         = ........days

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