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The water level just sits there even though the hose is going full blast.
(The barrel is made deep enough so that we don't have to worry about water overflowing the rim.) Henry Morris argued that if we started filling up our empty barrel it would take 30,000 years to reach the equilibrium point.
I will give you the setup of the problem and leave the algebra to you. We know at $t = 0$, we have 0\%$ of the Carbon, right? So then we know at $t = 5750$, there is \%$ of the carbon remaining. That gives you enough information to solve for $r$. Plug in your result for $r$ from the previous equation and solve for $t$.
This is why it is such a big concern when a nuclear submarine sinks... (By the way, you are mostly Carbon-12, which is not radioactive.
This argument was popularized by Henry Morris (1974, p.164), who used some calculations done in 1968 by Melvin Cook to get the 10,000-year figure. Whitelaw, using a greater ratio of carbon-14 production to decay, concluded that only 5000 years passed since carbon-14 started forming in the atmosphere!The water leaking out the sides of the barrel represents the loss (mainly by radioactive decay) of the atmosphere's supply of carbon-14.Now, the fuller that barrel gets the more water is going to leak out the thoroughly perforated sides, just as more carbon-14 will decay if you have more of it around.One of the most well-known applications of half-life is carbon-14 dating.The half-life of carbon-14 is approximately 5,730 years, and it can be reliably used to measure dates up to around 50,000 years ago.