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Answer :
The exponential decay model for a radioactive substance is N(t) =[tex]N₀e^(-t(ln(2)/39.4))[/tex].
To find how long it will take for a sample of 500 grams to decay to 400 grams, we can substitute N(t) = 400 and
N₀ = 500 into the decay model and solve for t, which is approximately 0.2236 years.
Radioactive decay is a process in which the number of atoms of a radioactive substance decreases over time. The exponential decay model for a radioactive substance is given by the formula N(t) = [tex]N₀e^(-kt)[/tex], where N(t) is the amount of substance at time t, N₀ is the initial amount of substance, k is the decay constant, and e is the base of the natural logarithm (approximately 2.71828).
In this case, the half-life of the substance is 39.4 years, which means that N₀/2 = [tex]N₀e^(-k*39.4)[/tex].
Solving this equation for k, we find k = ln(2)/39.4.
So, the exponential decay model for this substance is N(t) =[tex]N₀e^(-t(ln(2)/39.4))[/tex].
To find how long it will take for a sample of 500 grams to decay to 400 grams, we can substitute N(t) = 400 and
N₀ = 500 into the decay model and solve for t.
We get [tex]e^(-t(ln(2)/39.4))[/tex] = 400/500.
Taking the natural logarithm of both sides and solving for t, we find t ≈ 0.2236 years.
The third part of the question is incomplete.
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