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Samples of 2 radioactive isotopes, each with initial mass 1kg, are prepared in the Perplex Laboratory. Over time, the mass of the samples will decrease due to radioactive decay. The remaining mass, in kg, of each sample after t years can be modeled by the functions
where b>a>0.
Let M(t) be the combined mass (in kg) of the samples after t years.
Show that M(t)2−M(2t)=2⋅A(t)⋅B(t).
After 100 years, the combined mass remaining is 800g. After 200 years, the combined mass is 400g.
Using the identity in part (a), show that B(100)=25⋅A(100)3
Hence find A(100).
Find the exact value of a, giving your answer in the form klnp−lnq, where p,p,k∈Z.
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