Since radioactive material decays how is it possible that there is any left after 4.5 billion years? I was not able to find an answer for this question...
Some radioactive elements have half-life measured in thousands of years and some others even in millions, but over 4.5 billion years all the radioactive material that was part of the initial material that formed the planet earth should have decayed by now?
However, there is still radioactive material with short half-life to be found in nature. How is this possible and if the answer is that the new radioactive material is constantly being generated somehow, can you explain the mechanism of how this happens? 
Thanks.
 A: The half-life of Uranium 238 is about the age of the Earth, so only about half of the original supply should have decayed by now.  Also, there are some radioactive nuclei that get created by interactions with cosmic rays in the upper atmosphere (carbon-14) or decay from more stable nuclei (all of the daughter nuclei between U-238 and lead, for example).
A: It's because the half life time is also incredibly long.
The half-life of Uranium-238 is $4.5*10^9$ (=4.5 billion) years. Thorium-232 has $1.4*10^{10}$. Potassium-40 has $1.2*10^9$. These are all examples of primordial nuclides. Such half lives are of the order of the age of the universe.
There's also the effect of having a decay chain, since decay products themself can also be radioactive. Although if you look at the tables you'll find that only a few decay products have a significant lifetime -- most are either stable or negligble short. It does explain why you will always find materials with short half lives.
One noteworthy mentioning is ofcourse carbon-14, which is used in radiocarbon dating (i.e. estimating the life of certain soils or the remains of plants and animals). Here stable nitrogen-14 is turned into a radioactive carbon-14 by a collission with a cosmic neutron (which replaces one proton). The carbon is then absorbed from the atmosphere by plants or oceans. Carbon-14 has a half-life of roughly 6000 years, which is considrably shorter than the earths lifetime. 
A: Some elements with short half-lifes, are just decay products of those with long half-lifes.
A: The short half-life elements ocuring in nature come from the decay of long-half life elements. You can see examples of decay chains on this wikipedia page. For example, ²²⁴Ra (3.6 days half life) is produced by the decay of ²³²Th (14 billion years decay).
A: Since the universe has not yet run out of hydrogen, new stars are forming.  When those stars reach end of life and nova they produce the heavy elements beyond iron.
So until there are no more stars (large enough to create the heaviest elements such as uranium) and is no more hydrogen to form further stars, there will be a constant creation of further heavy elements.
A: Generally after 6 to 10 half lives radioactivity reduces very significantly. 133-In has half life of 180 milli secs (ms), so its radioactivity falls significantly after a few minutes. In comparison, 137-Cs with half life of 30.07 years show significantly low activity levels after 180 years or more.
A: Since radioactive materials are made in the intense power of super novas, radioactive materials can be created all the time since there are always supernovas going off and spreading this material around.
