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The problem provides a pendulum with its angular displacement given by the equation:

\[
\theta(t) = 0.18 \cos(9t)
\]

where:
- \(\theta(t)\) is the angular displacement in radians,
- \(t\) is time in seconds.

We need to determine:
1. The maximum angular displacement (\(\theta_{\text{max}}\)),
2. The rate of change of \(\theta\) (\(\theta'(t)\)) at \(t = 5\) seconds.

---

### 1. **Maximum Angular Displacement (\(\theta_{\text{max}}\))**

The maximum angular displacement is simply the amplitude of the cosine function. Since the amplitude of \(\cos(9t)\) is \(1\), multiplying it by \(0.18\) gives:

\[
\theta_{\text{max}} = 0.18 \, \text{radians}.
\]

---

### 2. **Rate of Change of \(\theta\) at \(t = 5\)**

To find \(\theta'(t)\), we differentiate \(\theta(t) = 0.18 \cos(9t)\) with respect to \(t\):

\[
\theta'(t) = -0.18 \cdot 9 \sin(9t).
\]

\[
\theta'(t) = -1.62 \sin(9t).
\]

At \(t = 5\):

\[
\theta'(5) = -1.62 \sin(9 \cdot 5).
\]

Simplify \(9 \cdot 5 = 45\), and find \(\sin(45)\):

\[
\sin(45) \approx 0.8509.
\]

So:

\[
\theta'(5) = -1.62 \cdot 0.8509 \approx -1.378 \, \text{rad/sec}.
\]

---

### Final Answers:
1. \(\theta_{\text{max}} = 0.180 \, \text{radians}\),
2. \(\theta'(5) = -1.378 \, \text{rad/sec}\).

Let me know if you'd like further clarifications or a step-by-step derivation. 

---

Here are 5 related questions for practice:
1. What is the period of the pendulum's oscillation based on the given equation?
2. How would \(\theta(t)\) change if the frequency were doubled?
3. What is the velocity of the pendulum when \(\theta(t)\) reaches its maximum?
4. How can you determine the maximum angular velocity of the pendulum?
5. How does the length of the pendulum (27 cm) affect its frequency?

**Tip:** Understanding the relationship between angular displacement, velocity, and acceleration in harmonic motion is key to solving pendulum problems!

A 27-centimeter pendulum moves according to the equation θ = 0.18 cos(9t), where θ is the angular displacement from the vertical in radians and t is the time in seconds. Determine the maximum angular displacement θmax and the rate of change of θ (in rad/sec) when t = 5 seconds. (Round your answers to three decimal places.)

Solve a pendulum motion problem involving the equation θ = 0.18 cos(9t). Learn how to find the maximum angular displacement (0.180 radians) and calculate the rate of change (θ'(5) ≈ -1.378 rad/sec) at t = 5 seconds using trigonometry and differentiation.

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