Sarvesh factory kept increasing its output by the same `%` every year. Find the `%` if it is known that his output is doubled after `2` years.

Options

A. `100sqrt(2)%`
B. `100(sqrt(2)-1)%`
C. `100(sqrt(2)+1)%`
D. `50(sqrt(2)+1)%`

Answer

To find the percentage increase in output per year for Sarvesh's factory, we can use the concept of compound interest. Here's the step-by-step solution:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We know that the output of the factory doubles in 2 years. This means if the initial output is represented as \( P \), after 2 years it becomes \( 2P \).

2. **Using the Compound Interest Formula**:
The formula for compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^n \]
Where:
- \( A \) is the amount after time \( n \),
- \( P \) is the principal amount (initial output),
- \( R \) is the rate of interest (percentage increase),
- \( n \) is the number of years.

In our case:
- \( A = 2P \) (since the output doubles),
- \( P = P \),
- \( n = 2 \).

3. **Setting Up the Equation**:
Plugging the values into the formula, we get:
\[ 2P = P \left(1 + \frac{R}{100}\right)^2 \]

4. **Simplifying the Equation**:
We can divide both sides by \( P \) (assuming \( P \neq 0 \)):
\[ 2 = \left(1 + \frac{R}{100}\right)^2 \]

5. **Taking the Square Root**:
To eliminate the square, we take the square root of both sides:
\[ \sqrt{2} = 1 + \frac{R}{100} \]

6. **Isolating \( R \)**:
Rearranging the equation to solve for \( R \):
\[ \frac{R}{100} = \sqrt{2} - 1 \]
\[ R = 100 \left(\sqrt{2} - 1\right) \]

7. **Calculating the Value of \( R \)**:
Now we can calculate \( R \):
- First, find \( \sqrt{2} \) which is approximately \( 1.414 \).
- Therefore, \( \sqrt{2} - 1 \approx 0.414 \).
- Now, multiply by 100:
\[ R \approx 100 \times 0.414 \approx 41.4 \]

8. **Conclusion**:
The percentage increase in output per year is approximately \( 41.4\% \).