The sample of rock is 3.52 million years old when the sample of rocks will have 15.625 grams of uranium -235
The half-life of a rock containing uranium -235 = 703.8 million years.
The initial quantity of uranium-235 in the given sample of rock = 500 grams.
We have to find after how many million years the sample of rock has 15.625 grams of uranium -235.
What is half life?
Half-life is the amount of time ( hours, days, years ) at which the value of any quantity is turned into half its initial value.
We have a formula for half-life:
[tex]N(t) = N_0 (\frac{1}{2})^{\frac{t}{t_{1/2}}}\\\\where~t_{\frac{1}{2}}= half-life ~of~ the ~quantity.\\\\N(t) = Final~quantity.\\\\N_0 = Initial ~quantity.[/tex]
We have,
Half-life of uranium -235 = 703.8
Initial amount of uranium -235 = 500 grams.
The final amount of uranium -235 = 15.625 grams.
So,
[tex]N(t) = N_0 (\frac{1}{2})^{\frac{t}{t_{1/2}}}\\\\15.625 = 500 (\frac{1}{2})^{\frac{t}{703.8}[/tex]
[tex]\frac{15.625}{500}=(\frac{1}{2})^{\frac{t}{703.8}}\\\\\frac{1}{32} = (\frac{1}{2})^{\frac{t}{703.8}}\\\\(\frac{1}{2} )^5= (\frac{1}{2})^{\frac{t}{703.8}}\\[/tex]
Since the base is the same we can equate the exponents.
5 = t / 703.8
t = 5 x 703.8
t = 3519 million years.
[ 1000 million years = 1 billion years ]
So,
3519 million years = ( 3519 / 1000 ) million years.
3519 million years = 3.519 million years.
Rounding to the nearest hundredths.
3519 million years = 3.52 million years.
Thus, the sample of rock will have 15.625 grams of uranium -235 in
3.52 million years.
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