The intensity of a light pulse travelling along a communication channel decreases exponetially with distance x according to the relation `I=I_(0)e^(-ax)` where `I_(0)` is the intensity at x=0 and `alpha` is the attenuation constant. The percentage decrease in intensity after a distance of `(("In"4)/(alpha))` is

Options

A. 0.7
B. 0.75
C. 0.8
D. 0.85

Answer

To solve the problem, we need to determine the percentage decrease in intensity after a distance of \( \frac{\ln 4}{\alpha} \) using the given exponential decay formula for intensity.

### Step-by-Step Solution:

1. **Write the given intensity equation**:
\[ I = I_0 e^{-\alpha x} \]
where \( I_0 \) is the initial intensity at \( x = 0 \) and \( \alpha \) is the attenuation constant.

**Hint**: Understand that this equation describes how intensity decreases with distance.

2. **Substitute the distance into the equation**:
We are interested in the intensity at \( x = \frac{\ln 4}{\alpha} \):
\[ I = I_0 e^{-\alpha \left(\frac{\ln 4}{\alpha}\right)} \]
Simplifying this gives:
\[ I = I_0 e^{-\ln 4} \]

**Hint**: Remember that \( e^{-\ln a} = \frac{1}{a} \).

3. **Simplify the expression**:
Using the property of logarithms:
\[ I = I_0 \cdot \frac{1}{4} = \frac{I_0}{4} \]

**Hint**: This shows that the intensity has decreased to one-fourth of its original value.

4. **Calculate the percentage decrease in intensity**:
The percentage decrease in intensity can be calculated using the formula:
\[ \text{Percentage Decrease} = \left( \frac{I_0 - I}{I_0} \right) \times 100 \]
Substituting \( I = \frac{I_0}{4} \):
\[ \text{Percentage Decrease} = \left( \frac{I_0 - \frac{I_0}{4}}{I_0} \right) \times 100 \]
Simplifying this:
\[ \text{Percentage Decrease} = \left( \frac{I_0 - \frac{I_0}{4}}{I_0} \right) \times 100 = \left( \frac{\frac{3I_0}{4}}{I_0} \right) \times 100 = \frac{3}{4} \times 100 = 75\% \]

**Hint**: The final result shows the proportion of intensity lost compared to the original intensity.

5. **Conclusion**:
The percentage decrease in intensity after a distance of \( \frac{\ln 4}{\alpha} \) is **75%**.