Calculate the half-life of a radioactive substance that decays by a factor of 3 in concentration every 1 million years

To calculate the half-life of a radioactive substance that decays by a factor of 3 in concentration every 1 million years, we can start by defining the concept of half-life.

The half-life is the amount of time it takes for half of the original quantity of a substance to decay. In this case, the substance decays by a factor of 3, so we can say that the half-life is the time it takes for the concentration to decrease to one-third (1/3) of its original value.

Now, if the substance decays by a factor of 3 every 1 million years, we can use this information to find the half-life. Let's work through the calculations step by step:

1. Start with an initial concentration of 1 unit.
2. After 1 million years, the concentration decreases by a factor of 3 to 1/3 units.
3. After another million years (a total of 2 million years), the concentration decreases by another factor of 3 to 1/3 * 1/3 = 1/9 units.
4. After three million years, the concentration decreases by another factor of 3 to 1/3 * 1/3 * 1/3 = 1/27 units.

At this point, it becomes evident that after each million years, the concentration should be divided by 3 again. So, after four million years, the concentration will be 1/81 units, and after five million years, it will be 1/243 units, and so on.

Continuing this pattern, we can see that it would take approximately 5 million years for the concentration to decrease to less than 1% of the original amount. Thus, we can conclude that the half-life of this radioactive substance is approximately 5 million years.