Ten thousand volunteers are registered with a charitable trust. The number of volunteers increases at the rate of 4% for every six months. Find the time period at the end of which the total number of volunteers becomes 10816.

Answer

To solve the problem of finding the time period at which the total number of volunteers becomes 10816, we can use the formula for compound interest since the number of volunteers increases at a certain percentage over time.

### Step-by-Step Solution:

1. **Identify the Initial Number of Volunteers (P)**:
The initial number of volunteers is given as:
\[ P = 10000 \]

2. **Identify the Rate of Increase (r)**:
The rate of increase is given as 4% every six months. We can express this as:
\[ r = 4\% = \frac{4}{100} = 0.04 \]

3. **Identify the Final Number of Volunteers (A)**:
The final number of volunteers we want to achieve is:
\[ A = 10816 \]

4. **Use the Compound Interest Formula**:
The formula for compound interest is:
\[ A = P(1 + r)^n \]
where \( n \) is the number of time periods (in this case, the number of 6-month intervals).

5. **Substitute the Known Values into the Formula**:
Plugging in the values we have:
\[ 10816 = 10000(1 + 0.04)^n \]

6. **Simplify the Equation**:
First, divide both sides by 10000:
\[ \frac{10816}{10000} = (1.04)^n \]
This simplifies to:
\[ 1.0816 = (1.04)^n \]

7. **Take the Logarithm of Both Sides**:
To solve for \( n \), we can take the logarithm of both sides:
\[ \log(1.0816) = n \cdot \log(1.04) \]

8. **Calculate the Logarithms**:
Using a calculator:
\[ \log(1.0816) \approx 0.0345 \]
\[ \log(1.04) \approx 0.0170 \]

9. **Solve for \( n \)**:
Now, divide both sides by \( \log(1.04) \):
\[ n = \frac{0.0345}{0.0170} \approx 2.03 \]

10. **Determine the Time Period**:
Since \( n \) represents the number of 6-month periods, we multiply by 6 to find the total time in months:
\[ \text{Total Time} = n \times 6 \approx 2.03 \times 6 \approx 12.18 \text{ months} \]

### Final Answer:
The time period at the end of which the total number of volunteers becomes 10816 is approximately **12 months**.