Find the percentage of disintegrated part of a radioactive element after 90 days, if `t_(1//2) = 30` days.

Options

A. 0.135
B. `46.5%`
C. `87.5%`
D. `90.15%`

Explanation

To find the percentage of the disintegrated part of a radioactive element after 90 days, given that the half-life \( t_{1/2} = 30 \) days, we can follow these steps:

### Step 1: Determine the number of half-lives
First, we need to calculate how many half-lives fit into the 90 days. The half-life is 30 days, so:

\[ \text{Number of half-lives} = \frac{\text{Total time}}{t_{1/2}} = \frac{90 \text{ days}}{30 \text{ days}} = 3 \]

### Step 2: Calculate the remaining quantity of the radioactive element
After each half-life, the quantity of the radioactive element is halved. If we start with an initial quantity \( N_0 \), after 3 half-lives, the remaining quantity \( N \) can be calculated as follows:

\[ N = N_0 \left( \frac{1}{2} \right)^3 = N_0 \left( \frac{1}{8} \right) \]

### Step 3: Calculate the disintegrated quantity
The disintegrated quantity is the initial quantity minus the remaining quantity:

\[ \text{Disintegrated quantity} = N_0 - N = N_0 - N_0 \left( \frac{1}{8} \right) = N_0 \left( 1 - \frac{1}{8} \right) = N_0 \left( \frac{7}{8} \right) \]

### Step 4: Calculate the percentage of disintegrated part
To find the percentage of the disintegrated part, we use the formula:

\[ \text{Percentage of disintegrated part} = \left( \frac{\text{Disintegrated quantity}}{N_0} \right) \times 100 = \left( \frac{N_0 \left( \frac{7}{8} \right)}{N_0} \right) \times 100 = \frac{7}{8} \times 100 \]

Calculating this gives:

\[ \frac{7}{8} \times 100 = 87.5\% \]

### Conclusion
Thus, the percentage of the disintegrated part of the radioactive element after 90 days is **87.5%**.

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