To find the average velocity of the particle over the given time intervals, we use the following formula for average velocity:

$\text{Average Velocity} = \frac{s(t_2) - s(t_1)}{t_2 - t_1}$

Where:

• $s(t)$ is the displacement function.
• $t_1$ and $t_2$ are the initial and final times of the interval.

Given $s(t) = 4\sin(\pi t) + 5\cos(\pi t)$, we calculate $s(t)$ at the required times.

(iii) Interval [1, 1.01]

1. Calculate $s(1)$: $s(1) = 4\sin(\pi \cdot 1) + 5\cos(\pi \cdot 1) = 4\sin(\pi) + 5\cos(\pi)$ Since $\sin(\pi) = 0$ and $\cos(\pi) = -1$, $s(1) = 4(0) + 5(-1) = -5 \text{ cm}$

2. Calculate $s(1.01)$: $s(1.01) = 4\sin(\pi \cdot 1.01) + 5\cos(\pi \cdot 1.01)$ $\sin(1.01\pi) \approx \sin(\pi + 0.01\pi) = -\sin(0.01\pi) \approx -0.0314$ $\cos(1.01\pi) \approx \cos(\pi + 0.01\pi) = -\cos(0.01\pi) \approx -0.9995$ $s(1.01) \approx 4(-0.0314) + 5(-0.9995) \approx -0.1256 - 4.9975 \approx -5.1231 \text{ cm}$

3. Average velocity over [1, 1.01]: $\text{Average Velocity} = \frac{-5.1231 - (-5)}{1.01 - 1} = \frac{-0.1231}{0.01} = -12.31 \text{ cm/s}$

Your original answer of $-12.31 \text{ cm/s}$ is actually correct. However, the issue may have been rounding errors or the specific interpretation by the software. Let’s check the exact rounding method.

(iv) Interval [1, 1.001]

1. Calculate $s(1.001)$: $s(1.001) = 4\sin(\pi \cdot 1.001) + 5\cos(\pi \cdot 1.001)$ $\sin(1.001\pi) \approx \sin(\pi + 0.001\pi) = -\sin(0.001\pi) \approx -0.00314$ $\cos(1.001\pi) \approx \cos(\pi + 0.001\pi) = -\cos(0.001\pi) \approx -0.999995$ $s(1.001) \approx 4(-0.00314) + 5(-0.999995) \approx -0.01256 - 4.999975 \approx -5.012535 \text{ cm}$

2. Average velocity over [1, 1.001]: $\text{Average Velocity} = \frac{-5.012535 - (-5)}{1.001 - 1} = \frac{-0.012535}{0.001} = -12.535 \text{ cm/s}$

However, if we round $-12.535 \text{ cm/s}$ to two decimal places: $\text{Average Velocity} \approx -12.54 \text{ cm/s}$

This result differs from your earlier answer, indicating that rounding or calculation precision may be the issue.

Summary:

• For part (iii) [1, 1.01], your answer $-12.31 \text{ cm/s}$ is indeed correct based on standard rounding.
• For part (iv) [1, 1.001], a more accurate answer after calculation and rounding is approximately $-12.54 \text{ cm/s}$.

Let me know if you need further details or have any questions.

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Further Questions:

1. How would the average velocity change if the time interval were significantly smaller?
2. What is the instantaneous velocity at $t = 1$?
3. How does the displacement function affect the velocity as time increases?
4. How would a phase shift in the trigonometric functions affect the velocity calculation?
5. What is the significance of rounding in such calculations?

Tip: When dealing with trigonometric functions, especially with small time intervals, small rounding errors can significantly affect the final answer. Use a high precision calculator for better accuracy.