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The rate of decay of a radioactive source falls from $3000$ counts per minute to $2000$ counts per minute 5 minutes later. From this information, determine the half life of the substance.

The answers for this question say to use the formula $N = Ae^{kt}$, using:

$A = 3000, N = 2000 \text { and } t = 50$ (to find $k$)

I don't understand how $3000$ can be used as an 'initial value', when it represents a rate of change? Wouldn't this be equal, then, to the derivative of the equation?

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$N = Ae^{kt}$ is a solution to the differential equation $$\frac{dN}{dt}=kN.$$ The question requires you to have the differential equation first. Initial value of $N$ is attained when $t=0 \iff N = Ae^{0}=A$, so $A$ is your initial value. Can you finish it?

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The differential equation in this case is $N'=kN$. The rate of change is $k$ and at this point it is unknown.

The initial count is $3000$ and the terminal count after $5$ minutes is $2000$ so you have $$2000=3000e^{5k}$$

You use logarithms to solve for $k$.

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