The position (in meters) of a particle moving on the x-axis is given by: ` x=2+9t +3t^(2) -t^(3) ,` where t is time in seconds . The distance travelled by the particle between t= 1s and t= 4s is m.

Explanation

To find the distance traveled by the particle between \( t = 1 \) s and \( t = 4 \) s, we will follow these steps:

### Step 1: Determine the position function
The position of the particle is given by:
\[ x(t) = 2 + 9t + 3t^2 - t^3 \]

### Step 2: Find the velocity function
To find the velocity, we differentiate the position function with respect to time \( t \):
\[ v(t) = \frac{dx}{dt} = \frac{d}{dt}(2 + 9t + 3t^2 - t^3) \]
Calculating the derivative:
\[ v(t) = 0 + 9 + 6t - 3t^2 = 9 + 6t - 3t^2 \]

### Step 3: Find when the velocity is zero
To find the points where the particle changes direction, we set the velocity function to zero:
\[ 9 + 6t - 3t^2 = 0 \]
Rearranging gives:
\[ -3t^2 + 6t + 9 = 0 \]
Dividing the entire equation by -3:
\[ t^2 - 2t - 3 = 0 \]
Factoring the quadratic:
\[ (t - 3)(t + 1) = 0 \]
Thus, the solutions are:
\[ t = 3 \quad \text{and} \quad t = -1 \]
Since time cannot be negative, we only consider \( t = 3 \).

### Step 4: Calculate the position at \( t = 1 \), \( t = 3 \), and \( t = 4 \)
Now we will calculate the position of the particle at these times.

1. **At \( t = 1 \)**:
\[ x(1) = 2 + 9(1) + 3(1^2) - (1^3) = 2 + 9 + 3 - 1 = 13 \, \text{m} \]

2. **At \( t = 3 \)**:
\[ x(3) = 2 + 9(3) + 3(3^2) - (3^3) = 2 + 27 + 27 - 27 = 29 \, \text{m} \]

3. **At \( t = 4 \)**:
\[ x(4) = 2 + 9(4) + 3(4^2) - (4^3) = 2 + 36 + 48 - 64 = 22 \, \text{m} \]

### Step 5: Calculate the distances
Now we can calculate the distances traveled in each segment:

1. **From \( t = 1 \) to \( t = 3 \)**:
\[ \text{Distance} = x(3) - x(1) = 29 - 13 = 16 \, \text{m} \]

2. **From \( t = 3 \) to \( t = 4 \)**:
\[ \text{Distance} = x(4) - x(3) = 22 - 29 = -7 \, \text{m} \]
Since distance cannot be negative, we take the absolute value:
\[ \text{Distance} = 7 \, \text{m} \]

### Step 6: Total distance traveled
The total distance traveled by the particle from \( t = 1 \) to \( t = 4 \) is:
\[ \text{Total Distance} = 16 + 7 = 23 \, \text{m} \]

### Final Answer
The distance traveled by the particle between \( t = 1 \) s and \( t = 4 \) s is \( \boxed{23} \) meters.

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