Contents

## Question:

The half-life of a radioactive substance is 3200 years. Find the quantity of the substance left at time if  ?

## Solution:

### Context

The general exponential decay function is defined as:  is the initial quantity, is the “proportionality constant”, is the initial time, and would be any time duration.

For radioactive decay problems, is treated as the “decay constant

Since , or the “half-life,” is the amount of time at which a radioactive material’s quantity is reduced to half, we can turn into,  where and .

We then solve for by canceling like terms and taking the natural logarithm of the equation: ((Recall that and ))  ### Actual solution

With these in mind, for Exercise 4.1.1, only algebra would be needed.

Given that and  , And since  , solving for would yield Therefore, the quantity over time of a 20 gram substance with a 3200 years half-life can be found using, Note:
We don’t simply use or as the solution because the resulting equation will NOT give us different values of at , only at . Remember that is just a value of .

Solution by Brian and Jian Hui

Print this page