Introduction:
The half-life of a radioactive substance is the time it takes for half of the original amount of the substance to decay. In this case, we are given that the half-life of cobalt 60 is 5.25 years.

Understanding the problem:
We are asked to determine the time it takes for the activity of a sample of cobalt 60 to decrease to about one third (1/3) of its original value. This means we need to find the time it takes for the amount of cobalt 60 to decrease to one third of its original amount.

Approach:
To solve this problem, we can use the concept of the half-life and exponential decay. Since we know the half-life of cobalt 60 is 5.25 years, we can use this information to find the time it takes for the activity to decrease to one third of its original value.

Solution:
Let's assume the original activity of the cobalt 60 sample is A₀.

1. Determine the number of half-lives:
To find the time it takes for the activity to decrease to one third of its original value, we need to determine the number of half-lives it takes for the activity to decrease by a factor of 1/3.

A decrease by a factor of 1/3 is equivalent to multiplying by 1/3. Therefore, the number of half-lives required can be found using the equation:

(1/2)^n = 1/3

where n is the number of half-lives.

2. Solve for n:
Taking the logarithm of both sides of the equation, we get:

n * log(1/2) = log(1/3)

Simplifying this equation, we find:

n = log(1/3) / log(1/2)

Using the properties of logarithms, we can evaluate this expression to find the value of n.

3. Calculate the time:
Since each half-life is 5.25 years, we can calculate the total time it takes for the activity to decrease to one third of its original value by multiplying the number of half-lives (n) by the half-life period (5.25 years).

Thus, the time it takes for the activity to decrease to about one third (1/3) of its original value is approximately 8.3 years.

Conclusion:
The time it takes for the activity of a sample of cobalt 60 to decrease to about one third (1/3) of its original value is approximately 8.3 years. This can be calculated by determining the number of half-lives required for the activity to decrease by a factor of 1/3 and then multiplying that number by the half-life period of cobalt 60.