Question

Problem PageQuestion The mass of a radioactive substance follows a continuous exponential decay model, with a decay rate parameter of per day. Find the half-life of this substance (that is, the time it takes for one-half the original amount in a given sample of this substance to decay). Note: This is a continuous exponential decay model. Do not round any intermediate computations, and round your answer to the nearest hundredth.

Answer 1

Answer:

t = 1.51

Step-by-step explanation:

Exponential Model

The exponential model is often used to simulate the behavior of a magnitude that either grow or decay in proportion to the existing amount of that magnitude.

The model can be expressed as

$M=M_oe^{kt}$

In this case, Mo is the initial mass of the radioactive substance and k is a constant which value is positive if the mass is growing or negative if the mass is decaying.

The value of k is not precisely given in the question, we are assuming $k=-0.2$

The model is now

$M=M_oe^{-0.2t}$

We are required to compute the time it takes the mass to reach one-half of its initial value:

$\displaystyle \frac{M_o}{2}=M_oe^{-0.2t}$

Simplifying

$\displaystyle \frac{1}{2}=e^{-0.2t}$

Taking logarithms

$\displaystyle ln\frac{1}{2}=ln(e^{-0.2t})=-0.2t$

Solving for t

$\displaystyle t=-\frac{ln\frac{1}{2}}{0.2}=1.51$