Question

question an unknown radioactive element decays into non - radioactive substances. in 30 days the radioactivity of a sample decreases by 12%. the exponential decay model for the decay after t days is given by $p(t)=p_0e^{-0.004261t}$. when will a sample of 50 mg decay to 10 mg? round your final answer to 1 decimal place. provide your answer below:

Explanation:

Step1: Substitute values into the decay formula

We are given $P(t)=10$, $P_0 = 50$ and $P(t)=P_0e^{- 0.004261t}$. So, $10 = 50e^{-0.004261t}$.

Step2: Simplify the equation

Divide both sides of the equation by 50: $\frac{10}{50}=e^{-0.004261t}$, which simplifies to $0.2 = e^{-0.004261t}$.

Step3: Take the natural - logarithm of both sides

$\ln(0.2)=\ln(e^{-0.004261t})$. Since $\ln(e^x)=x$, the right - hand side becomes $-0.004261t$. So, $\ln(0.2)=-0.004261t$.

Step4: Solve for $t$

We know that $\ln(0.2)\approx - 1.60944$. Then $t=\frac{\ln(0.2)}{-0.004261}=\frac{-1.60944}{-0.004261}\approx377.7$.

Answer:

$377.7$