Question

If the half-life of a radioactive material is 10 years, then the percentage of the material decayed in 30 years is

• A. 87.5
• B. 78.5
• C. 58.7
• D. 48

Hint:

The percentage of radioactive material decayed after $n$ half-lives is $\left( 1 - \left( \frac{1}{2} \right)^n \right) \times 100$, where $n = \frac{t}{T}$.

Answer 1

The Correct Option is A

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 $t$ is given by:
\[ N = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T}} \]
where:

• $N_0$ is the initial amount,
• $N$ is the amount remaining after time $t$,
• $T$ is the half-life,
• $t$ is the elapsed time.

The percentage decayed is:
\[ \text{Percentage decayed} = \left( 1 - \frac{N}{N_0} \right) \times 100 \]
Step 2: Identify the given values and calculate the fraction remaining

• Half-life, $T = 10 \, \text{years}$
• Time, $t = 30 \, \text{years}$

Number of half-lives:
\[ \frac{t}{T} = \frac{30}{10} = 3 \]
Fraction remaining:
\[ \frac{N}{N_0} = \left( \frac{1}{2} \right)^3 = \frac{1}{8} \]
Percentage decayed:
\[ \text{Percentage decayed} = \left( 1 - \frac{1}{8} \right) \times 100 = \frac{7}{8} \times 100 = 87.5% \]
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.