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Moon rocks contain small amounts of uranium-238, which decays into lead with a half-life of 4.5 billion years. In one rock, you determine that 60% of the original uranium-238 remains, while the other 40% has decayed into lead. How old is the rock?

Answer 1

Lydia Allen

3.3 billion years,

60% of both rocks are still present.

After T billion years, if m% of the mass is still present,

#m=ce^(-kT)#, where c and k are constants for the radioactive material that is decaying.,

T = 4.5 when m = 50 and 100 when T = 0.

So, #c = 100 and 50 = 100 e^(-4.5k)#.

So, #e^(-4.5k)=1/2 and k = ln 2/4.5=0.1540#.

Now,# m=100e^(-0.154 T)# Inversely, #T=ln(100/m)/0.154#. .

For the rock that had lost 40%, 60% remains. Its age T is given by #T = ln(100/60)/0.154=3.3# billion years, nearly.

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Answer 2

Nova Adams

The rock is approximately 1.5 billion years old.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.