The half-life of a radioactive substance is `30` minutes, The time (in minutes) taken between `40 %` decay and `85 %` decay of the same radioactive substance is.

Options

A. 15
B. 30
C. 45
D. 60

Explanation

To solve the problem of finding the time taken between 40% decay and 85% decay of a radioactive substance with a half-life of 30 minutes, we can follow these steps:

### Step 1: Understand the Decay Process
The half-life of a radioactive substance is the time required for half of the substance to decay. In this case, the half-life is given as 30 minutes.

### Step 2: Determine Remaining Substance After Decay
Let the initial amount of the substance be \( N_0 \).

- After 40% decay, 60% of the substance remains:
\[ N_1 = N_0 - 0.4N_0 = 0.6N_0 \]

- After 85% decay, 15% of the substance remains:
\[ N_2 = N_0 - 0.85N_0 = 0.15N_0 \]

### Step 3: Find the Ratio of Remaining Substances
Now, we can find the ratio of the remaining substances after 40% and 85% decay:
\[ \frac{N_2}{N_1} = \frac{0.15N_0}{0.6N_0} = \frac{0.15}{0.6} = \frac{1}{4} \]

### Step 4: Relate the Ratio to Half-Lives
The ratio \( \frac{1}{4} \) can be expressed in terms of half-lives:
\[ \frac{1}{4} = \left(\frac{1}{2}\right)^2 \]
This indicates that the decay from \( N_1 \) to \( N_2 \) corresponds to 2 half-lives.

### Step 5: Calculate the Total Time Taken
Since each half-life is 30 minutes, the total time taken for 2 half-lives is:
\[ \text{Total time} = 2 \times \text{half-life} = 2 \times 30 \text{ minutes} = 60 \text{ minutes} \]

### Final Answer
The time taken between 40% decay and 85% decay is **60 minutes**.

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