A graphing calculator is recommended.
The displacement (in centimeters) of a particle s moving back and forth along a straight line is given by the equation 
s = 4 sin(𝜋t) + 5 cos(𝜋t),
 where t is measured in seconds. (Round your answers to two decimal places.)
(a)
Find the average velocity during each time period.
(i)
[1, 2]
10.00
 
Correct: Your answer is correct.
 cm/s
(ii)
[1, 1.1]
-9.91
 
Correct: Your answer is correct.
 cm/s
(iii)
[1, 1.01]
-12.31
 
Incorrect: Your answer is incorrect.
 cm/s
(iv)
[1, 1.001]
-12.38
 
Changed: Your submitted answer was incorrect. Your current answer has not been submitted.
 cm/s

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.

---

**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.

A graphing calculator is recommended. The displacement (in centimeters) of a particle moving back and forth along a straight line is given by the equation s = 4 sin(𝜋t) + 5 cos(𝜋t), where t is measured in seconds. (Round your answers to two decimal places.) (a) Find the average velocity during each time period. (i) [1, 2] (ii) [1, 1.1] (iii) [1, 1.01] (iv) [1, 1.001]

Explore how to calculate the average velocity of a particle moving in a straight line with displacement described by trigonometric functions. This problem includes step-by-step solutions using the formula s(t) = 4 sin(πt) + 5 cos(πt), and covers key concepts like trigonometry and average velocity. Suitable for advanced high school students.

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