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We are given the displacement function \( s(t) \), which describes the displacement (in centimeters) of a particle from an origin \( O \) at time \( t \) seconds as follows:

\[
s(t) = t^3 \sin t + 3t^2 \cos t, \quad 0 \leq t \leq 2.5.
\]

We are tasked with finding the maximum distance of the particle from \( O \). To solve this:

### Steps:
1. **Differentiate the displacement function** \( s(t) \) with respect to \( t \), and find the critical points by setting the derivative equal to zero.
2. **Evaluate** \( s'(t) = 0 \) to find the values of \( t \) where local extrema might occur.
3. **Check the endpoints** and the critical points to determine the maximum value of \( s(t) \) on the interval \( [0, 2.5] \).

Let's start by differentiating \( s(t) \).The derivative of the displacement function \( s(t) \) is:

\[
s'(t) = t^3 \cos t + 6t \cos t.
\]

Setting this equal to zero to find the critical points yields the following potential solutions:

\[
t = 0, \ \frac{\pi}{2}, \ \frac{3\pi}{2}, \ \pm i\sqrt{6}.
\]

However, the imaginary solutions \( \pm i\sqrt{6} \) are not physically relevant since we are only considering real values of \( t \) within the interval \( [0, 2.5] \). Therefore, we need to evaluate:

\[
t = 0, \ \frac{\pi}{2} \ (\approx 1.57), \ \frac{3\pi}{2} \ (\approx 4.71).
\]

Since \( \frac{3\pi}{2} \) is outside the interval \( [0, 2.5] \), we disregard it, leaving \( t = 0 \) and \( t = \frac{\pi}{2} \) as critical points. Now, we will evaluate the displacement function \( s(t) \) at \( t = 0, \ t = \frac{\pi}{2}, \) and the endpoint \( t = 2.5 \) to find the maximum value.The values of the displacement function \( s(t) \) at the critical points and the endpoint are:

- \( s(0) = 0 \)
- \( s\left(\frac{\pi}{2}\right) \approx 3.88 \)
- \( s(2.5) \approx -5.67 \)

Therefore, the maximum displacement occurs at \( t = \frac{\pi}{2} \), with the maximum distance from the origin being approximately **3.88 cm**.

Would you like more details or have any questions about the process?

Here are five related questions you might consider:
1. How do you verify whether a critical point gives a maximum or minimum?
2. What would the motion of the particle look like graphically over time?
3. How do higher derivatives help in analyzing the nature of critical points?
4. How would the result change if the time interval were extended beyond 2.5 seconds?
5. What is the significance of evaluating endpoints in optimization problems?

**Tip:** When finding maxima and minima, always check the values at critical points and endpoints to ensure you identify the true extremum over the given interval.

Write down the maximum distance of the particle from O given the displacement function s(t) = t^3 sin(t) + 3t^2 cos(t) over the interval 0 ≤ t ≤ 2.5.

Discover how to find the maximum distance of a particle from the origin using the displacement function s(t) = t^3 sin(t) + 3t^2 cos(t) over the interval 0 ≤ t ≤ 2.5. Learn the steps involving differentiation, identifying critical points, and evaluating endpoints for optimization. Suitable for Grades 11-12 students.

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