Answer: A) 6
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Explanation:
The half-life formula is
y = A*(1/2)^(x/H)
where A is the starting amount, H is the half-life, x is the time value and y is the amount after x units of time have passed.
I'll express the x and H values in terms of millions of years. So for example, when we say H = 500, then that really refers to 500 million years.
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In this case, we know that,
Let's use those values to solve for x. We'll need logs to cut the exponents down from the trees.
y = A*(1/2)^(x/H)
3 = 192*(1/2)^(x/500)
3/192 = (0.5)^(x/500)
0.015625 = (0.5)^(x/500)
log(0.015625) = log( (0.5)^(x/500) )
log(0.015625) = (x/500)*log(0.5)
(x/500)*log(0.5) = log(0.015625)
x = 500*log(0.015625)/log(0.5)
x = 3,000
Keep in mind that x is in millions of years. So saying x = 3,000 means 3,000 million years. This translates to 3,000*10^6 = 3,000,000,000 = 3 billion years.
It will take about 3 billion years for the 192 atoms to decay to only 3 atoms.
The amount of half-lives this occurs is x/H = (3,000)/500 = 6
So the time span of 3 billion years is the same as 6 half-lives when each half-life is 500 million years long.
