Question 165711This question is from textbook
: The level of thorium in a sample decreases by a factor of one half every 4.2 million years. A meteorite is discovered to have only 7.6% of its orginial thorium remaing. How old is the meteorite?
This question is from textbook
Found 2 solutions by scott8148, gonzo:
Answer by scott8148(6628)   (Show Source):
You can put this solution on YOUR website! you start with 100% (1.00)
after 4.2 million years, you have 50% (.50) or 1 * .5 or (.5)^1
after another 4.2 million years, you have 25% (.25) or 1 * .5 * .5 or (.5)^2
after N cycles (they're called half-lives) you have (.5)^N remaining
(.5)^N=.076 __ taking log __ N(log(.5))=log(.076) __ dividing by log(.5) __ N=(log(.076))/(log(.5))
using a calculator __ N=3.7 (approx)
so the age of the meteorite is N times 4.2 million or 15.5 million (approx)
Answer by gonzo(654)  (Show Source):
You can put this solution on YOUR website! let x = number of half lives.
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answer is x = 3.717856771 half lives.
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3.717856771 half lives * 4.2 million years = 15.61499844 million years old.
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a rough measure proves this to be correct.
example:
100% / 2 = 50% / 2 = 25% / 2 = 12.5% / 2 = 6.25%
the answer in half lives has to be between 3 and 4.
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the exact calculation is done as follows:
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the equation for half lives is (1/2)^x where x becomes the number of half lives.
example:
(1/2)^1 = 1 half life
(1/2)^2 = 2 half lives
(1/2)^3 = 3 half lives
etc.
100 * 1/2 = 50 * 1/2 = 25 * 1/2 = 12.5, etc.
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100% went to 7.6%
equation is therefore
100% * (1/2)^x = 7.6%
divide both sides of the equation by 100% and it becomes:
1*(1/2)^x = .076
which becomes:
(1/2)^x = .076
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i used logarithms to solve this.
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the general form of the logarithmic equation is:
a^x = y if and only if log_a (y) = x
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i use log_a to mean log subscript a which i can't do here.
it means logarithm to the base a.
just like log_2 = log to the base 2
log_10 = log to the base 10, or just log (10 is implied if the base is not shown).
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in the equation (1/2)^x = .076,
the base is (1/2) = .5
the exponent is x
y = .076
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.076 = (1/2)^x if and only if log_.5 (.076) = x
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log_.5 (.076) = x (equation 1)
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in order to use the calculator to solve this, i need the conversion formula to get a logarithm from any base to the base 10 (or ln) which the calculator can use.
i'll work with the base 10.
the conversion formula is as follows:
log_base (any number) = log_10 (any number) / log_10 (base)
as an example:
take 5^x = 390625
logarithmic form of this would be: log_5 (390625) = x
i can't use the formula as is because the calculator won't handle bases other than 10 or e.
i wish to convert to base 10.
log_5 (390625) = log_10 (390625) / log_10 (5) = 8
answer is 8 which i know to be correct because that's how i set it up.
if i used base e i would take the natural log, called ln.
log_5 (390625) = ln (390625) / ln (5) = 8.
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anyway, using the calculator, i solved the equation as follows:
log_.5 (.076) = x (equation 1)
converting to base 10, equation becomes:
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log_10 (.076) / log_10 (.5) = x
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x = 3.717856771 = number of half lives to get from 100 to 7.6.
this is the same as saying it's the number of times the value is halved.
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