A partical moves in a straight line with a velocity given by `(dx)/(dt)=x+1` (x is the distance described). The time taken by a particle to traverse a distance of 99 metres, is

Answer

To solve the problem, we need to find the time taken by a particle to traverse a distance of 99 meters, given that its velocity is expressed as \(\frac{dx}{dt} = x + 1\).

### Step-by-Step Solution:

1. **Write the given equation:**
\[ \frac{dx}{dt} = x + 1 \]

2. **Separate the variables:**
We can rearrange the equation to separate the variables \(x\) and \(t\):
\[ \frac{dx}{x + 1} = dt \]

3. **Integrate both sides:**
Now, we integrate both sides:
\[ \int \frac{dx}{x + 1} = \int dt \]
The left side integrates to:
\[ \ln |x + 1| + C_1 \]
and the right side integrates to:
\[ t + C_2 \]
Therefore, we can combine the constants:
\[ \ln |x + 1| = t + C \]

4. **Solve for the constant \(C\):**
To find the constant \(C\), we use the initial condition. Assume that at \(t = 0\), \(x = 0\):
\[ \ln |0 + 1| = 0 + C \implies \ln 1 = C \implies C = 0 \]
Thus, the equation simplifies to:
\[ \ln |x + 1| = t \]

5. **Substitute \(x = 99\) to find \(t\):**
Now, we want to find the time \(t\) when \(x = 99\):
\[ \ln |99 + 1| = t \implies \ln 100 = t \]

6. **Simplify \(t\):**
We can express \(100\) as \(10^2\):
\[ t = \ln(10^2) = 2 \ln 10 \]

7. **Final answer:**
Therefore, the time taken by the particle to traverse a distance of 99 meters is:
\[ t = 2 \ln 10 \text{ seconds} \]