Radioactive Decay

The relationship between the mean life time and the half-life time for a radioactive substance is given by the following formula: This formula can be derived from the basic concepts of radioactive decay. The mean lifetime is related to the decay constant by , and similarly, the half-life is related to by . Therefore, combining these two gives the final result:

The relation between the amount of a radioactive element at time and its half-life is given by the formula: where is the initial number of atoms, is the half-life, and is the number of atoms at time . The ratio implies that 4 half-lives have passed.
Thus, hours.

Let’s break this down step by step to calculate the percentage of the radioactive material decayed in 30 years and determine why option (1) is the correct answer.
Step 1: Understand radioactive decay and the half-life formula
The amount of radioactive material remaining after time is given by:

where:

• is the initial amount,
• is the amount remaining after time ,
• is the half-life,
• is the elapsed time.

The percentage decayed is:

Step 2: Identify the given values and calculate the fraction remaining

• Half-life,
• Time,

Number of half-lives:

Fraction remaining:

Percentage decayed:

Step 3: Confirm the correct answer
The calculated percentage decayed is 87.5%, which matches option (1).
Thus, the correct answer is Jonn of 87.5.

Step 1: Use the formula for radioactive decay.
Step 2: Given and min, solve for : Step 3: Use half-life formula : Step 4: Calculate numerically: Step 5: Select the correct option.
The half-life of the substance is approximately 6 minutes, matching option (4).