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 \( t = 1 \) second and \( t = 4 \) seconds, we will follow these steps:

### Step 1: Determine the position of the particle at \( t = 1 \) second.
Given the position function:
\[ x(t) = 2 + 9t + 3t^2 - t^3 \]
Substituting \( t = 1 \):
\[ x(1) = 2 + 9(1) + 3(1^2) - (1^3) \]
Calculating this:
\[ x(1) = 2 + 9 + 3 - 1 = 13 \text{ meters} \]

### Step 2: Determine the position of the particle at \( t = 4 \) seconds.
Now substituting \( t = 4 \):
\[ x(4) = 2 + 9(4) + 3(4^2) - (4^3) \]
Calculating this:
\[ x(4) = 2 + 36 + 3(16) - 64 \]
\[ x(4) = 2 + 36 + 48 - 64 \]
\[ x(4) = 2 + 36 + 48 - 64 = 22 \text{ meters} \]

### Step 3: Calculate the distance traveled between \( t = 1 \) and \( t = 4 \).
The distance traveled is given by:
\[ \text{Distance} = x(4) - x(1) \]
Substituting the values we calculated:
\[ \text{Distance} = 22 - 13 = 9 \text{ meters} \]

### Final Answer:
The distance traveled by the particle between \( t = 1 \) second and \( t = 4 \) seconds is \( 9 \) meters.

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