## Calcule a meia-vida e a quantidade de uma substância passo a passo

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 $N(t)=N_0e^{-\lambda t}$ 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 $\lambda$ (also known as decay constant or decay rate).

We know that after half-life there will be twice less the initial quantity: $N\left(t_h\right)=\frac{N_0}{2}=N_0e^{-\lambda t_h}$.

Simplifying gives $\frac{1}{2}=e^{-\lambda t_h}$ or $\lambda=-\frac{\ln\left(\frac{1}{2}\right)}{t_h}$.

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: $N(t)=\frac{125 \sqrt[3]{2}}{64}\approx 2.46078330057592$.