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We are given the position function of an object moving along a horizontal path:

\[
x(t) = -t^2 + 4t - 2
\]

where \(x(t)\) represents the position in meters, and \(t\) is time in seconds. The problem asks for the **distance traveled** by the object during the time interval \(0 \leq t \leq 5\).

### Solution Outline

1. **Velocity function**: The velocity is the derivative of the position function:
   \[
   v(t) = \frac{dx(t)}{dt} = -2t + 4
   \]

2. **Identify when velocity changes sign**: The object’s direction changes when \(v(t) = 0\), which occurs at:
   \[
   -2t + 4 = 0 \implies t = 2 \text{ seconds}
   \]
   So, the object changes direction at \(t = 2\).

3. **Calculate positions at critical points**: We need to evaluate the position function \(x(t)\) at \(t = 0\), \(t = 2\), and \(t = 5\).

   - \(x(0) = -(0)^2 + 4(0) - 2 = -2\)
   - \(x(2) = -(2)^2 + 4(2) - 2 = 2\)
   - \(x(5) = -(5)^2 + 4(5) - 2 = -7\)

4. **Total distance traveled**: Since the object changes direction at \(t = 2\), we break the distance into two parts:
   - Distance from \(t = 0\) to \(t = 2\): 
     \[
     |x(2) - x(0)| = |2 - (-2)| = 4
     \]
   - Distance from \(t = 2\) to \(t = 5\): 
     \[
     |x(5) - x(2)| = |-7 - 2| = 9
     \]

   The total distance traveled is:
   \[
   4 + 9 = 13 \text{ meters}
   \]

### Final Answer:
The object travels **13 meters** in total during the interval \(0 \leq t \leq 5\).

Would you like more details, or do you have any questions?

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Here are five follow-up questions related to this problem:
1. How would the total distance change if the time interval were \(0 \leq t \leq 6\)?
2. What is the object's velocity at \(t = 1\) and \(t = 4\)?
3. How can we find the total displacement instead of the total distance?
4. What would happen to the object's motion if the equation of \(x(t)\) were changed to \(x(t) = -t^2 + 6t - 5\)?
5. How is the concept of total distance traveled different from displacement in physics?

Tip: The total distance is always the sum of the absolute values of changes in position, even if the object changes direction.

The motion of an object along a horizontal path is modeled by the equation x(t) = -t^2 + 4t - 2, where x(t) is the position from the origin in meters and t is the time in seconds. Find the distance traveled by the object on the time interval 0 ≤ t ≤ 5.

Explore how to find the total distance traveled by an object along a horizontal path using calculus. Given the position function x(t) = -t^2 + 4t - 2, the solution includes calculating velocity, identifying changes in direction, and summing absolute distances over the interval 0 ≤ t ≤ 5.

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