Correct Answer - Option 3 : 2 ln 5
Formula Used:
\(\int \frac {1}{x} dx = lnx + c\)
At t = 0, Distance covered (x) = 0
Distance covered by the particle (x) = 24 m
Calculation:
The given differential equation \(\rm \frac{dx}{dt}=x+1\) is in variable separable form.
On separating the variables, we get
⇒ \(\rm \left(\frac{1}{x+1}\right)dx = dt\)
On integrating both sides, we get:
⇒ ln (x + 1) = t + C ---(i)
According to the question at t = 0, x = 0.
⇒ ln (0 + 1) = 0 + C
⇒ C = 0
Now, From (i), we get
⇒ ln (x + 1) = t
Again according to the question
⇒ t = ln (24 + 1)
⇒ t = ln 25
⇒ t = ln 52
∴ The required time is 2 ln 5.