# Half-Life Calculator

This calculator will calculate the half-life, initial quantity, quantity remained and time, with steps shown.

There are units of mass of a substance with a half-life of units of time. In units of time, there will remain units of mass of the substance.

Enter any three values.

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## Solution

Your input: find $$N(t)$$$in $$N(t)=N_0e^{-kt}$$$ given $$N_0=250$$$, $$t_h=15$$$, $$t=100$$$. $$N(t)$$$ is the amount after the time $$t$$$, $$N_0$$$ is the initial amount, $$t_h$$$is the half-life. First, find the constant $$k$$$.

We know that after half-life there will be twice less the initial quantity: $$N\left(t_h\right)=\frac{N_0}{2}=N_0e^{-k t_h}$$$. Simplifying gives $$\frac{1}{2}=e^{-k t_h}$$$ or $$k=-\frac{\ln\left(\frac{1}{2}\right)}{t_h}$$$. Plugging this into the initial equation, we obtain that $$N(t)=N_0e^{\frac{\ln\left(\frac{1}{2}\right)}{t_h}t}$$$ or $$N(t)=N_0\left(\frac{1}{2}\right)^{\frac{t}{t_h}}$$$. Finally, just plug in the given values and find the unknown one. From $$N(t)=250\left(\frac{1}{2}\right)^{\frac{100}{15}}$$$, we have that $$N(t)=\frac{125 \sqrt[3]{2}}{64}$$$. Answer: $$N(t)=\frac{125 \sqrt[3]{2}}{64}\approx 2.46078330057592$$$.