Age of the Universe: Solve for t_0

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To solve for t_0 in a flat universe, the relationship t_0 = (2/3)(H_0)^(-1) needs to be established. The discussion emphasizes the importance of Hubble's law, which relates velocity and distance in an expanding universe. The user expresses confusion about how to apply the concept of a flat universe and the implications of a constant Hubble constant. They consider using kinematic equations but are unsure how to proceed with their calculations. Guidance is sought on the next steps to derive the solution effectively.
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Hi all,
Hoping you can help with this problem!

Homework Statement


For a flat universe, show that t_{0}=2/3 (H_{0})^{-1} ? Seems like a simple problem I just don't really know where to start.

Homework Equations


Hubble's law v= H_{0}r
and then t= 1/H_{0}

The Attempt at a Solution


I just need guidance where to start. Flat universe means that expansion is constant in any direction right? What do I do?
 
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flat universe means the Hubble constant is constant.

Then 1/hubble constant has units time.
Think of the equations for something that doubles/halves with a constant time interval
 
How about the kinematic equation: x(t)= vt + 1/2at^2
Not sure if this is right and what to do with it? maybe take a derivatives of both sides?
 
I still can't figure out the answer, can someone give me a hint please
 

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