A particle moves along a horizontal line such that its position at time t is specified by s(t) = t^3 - 12t^2 + 36t - 30. What is the total distance travelled by the particle?
Understand the Problem
The question is asking for the total distance traveled by a particle whose position is described by the function s(t) = t^3 - 12t^2 + 36t - 30. To solve this, we would analyze the motion over the duration specified and calculate the total distance based on the given position function.
Answer
The total distance traveled by the particle is $96 \text{ ft}$.
Answer for screen readers
The total distance traveled by the particle is $96 \text{ ft}$.
Steps to Solve
- Find the velocity function
To find the velocity of the particle, we need to differentiate the position function $s(t)$.
$$ v(t) = \frac{ds}{dt} = 3t^2 - 24t + 36 $$
- Determine when the particle changes direction
To find the points where the particle changes direction, set the velocity function equal to zero and solve for $t$:
$$ 3t^2 - 24t + 36 = 0 $$
Dividing the equation by 3 gives:
$$ t^2 - 8t + 12 = 0 $$
Now, factor the quadratic:
$$(t - 6)(t - 2) = 0$$
Thus, the critical points are $t = 2$ and $t = 6$.
- Evaluate position at critical points and endpoints
We need to evaluate the position function at the critical points and also at $t = 0$ (if specified) to find the positions:
- $s(0) = 0^3 - 12(0)^2 + 36(0) - 30 = -30$
- $s(2) = 2^3 - 12(2)^2 + 36(2) - 30 = 18$
- $s(6) = 6^3 - 12(6)^2 + 36(6) - 30 = -30$
- Calculate total distance traveled
Now, we calculate the total distance traveled by taking the absolute differences in position between the critical points:
-
From $t = 0$ to $t = 2$: $$ |s(2) - s(0)| = |18 - (-30)| = 48$$
-
From $t = 2$ to $t = 6$: $$ |s(6) - s(2)| = |-30 - 18| = 48$$
- Add the distances together
Finally, add together the absolute distances computed:
$$ \text{Total distance} = 48 + 48 = 96$$
Therefore, if the time duration is limited to a specific range, ensure to adjust calculations based on that.
The total distance traveled by the particle is $96 \text{ ft}$.
More Information
This particle motion involves polynomial functions, and we analyzed its movement by finding velocity, direction changes, and evaluating position at critical points to compute total distance.
Tips
- Failing to differentiate the position function correctly.
- Not considering the absolute distances when the particle changes direction.
- Ignoring evaluation at the endpoints when applicable.
AI-generated content may contain errors. Please verify critical information