A particle is travelling along a straight line OX. The distance r of the particle from O at a timet is given by x = 37 + 27t- `t^(3)`, where t is time in seconds. The distance of the particle from O when it comes to rest is

Options

A. 81 m
B. 91 m
C. 101 m
D. 111m

Explanation

To solve the problem, we need to find the distance of the particle from point O when it comes to rest. Here are the steps to arrive at the solution:

### Step-by-Step Solution:

1. **Understand the given equation**: The distance \( x \) of the particle from point O at time \( t \) is given by the equation:
\[ x(t) = 37 + 27t - t^3 \]

2. **Find the velocity**: The velocity \( v \) of the particle is the derivative of the distance with respect to time. We differentiate \( x(t) \):
\[ v(t) = \frac{dx}{dt} = \frac{d}{dt}(37 + 27t - t^3) = 27 - 3t^2 \]

3. **Set the velocity to zero**: To find when the particle comes to rest, we set the velocity \( v(t) \) to zero:
\[ 27 - 3t^2 = 0 \]

4. **Solve for \( t \)**: Rearranging the equation gives:
\[ 3t^2 = 27 \implies t^2 = 9 \implies t = 3 \text{ or } t = -3 \]
Since time cannot be negative, we take \( t = 3 \) seconds.

5. **Substitute \( t \) back into the distance equation**: Now we substitute \( t = 3 \) back into the original distance equation to find the distance from point O:
\[ x(3) = 37 + 27(3) - (3)^3 \]
Calculating this step-by-step:
- Calculate \( 27(3) = 81 \)
- Calculate \( (3)^3 = 27 \)
- Substitute these values into the equation:
\[ x(3) = 37 + 81 - 27 \]
- Now, simplify:
\[ x(3) = 37 + 81 = 118 - 27 = 91 \]

6. **Final answer**: The distance of the particle from point O when it comes to rest is:
\[ \boxed{91 \text{ meters}} \]