To find the value of $\theta$ in the interval $0 \leq \theta \leq \frac{\pi}{2}$ that corresponds to the point on the curve $r = -4 - \sin(4\theta) - \theta \cos(\theta)$ with the shortest distance from the pole (origin), we need to find the minimum value of the function $r(\theta)$ within the given interval. The distance from the pole is minimized when $r(\theta)$ is minimized.

Steps to solve:

1. Define the function: $r(\theta) = -4 - \sin(4\theta) - \theta \cos(\theta)$

2. Take the derivative of $r(\theta)$ with respect to $\theta$ to find the critical points: $\frac{dr}{d\theta} = -4 \cos(4\theta) - \cos(\theta) + \theta \sin(\theta)$

3. Set the derivative equal to zero to find critical points: $-4 \cos(4\theta) - \cos(\theta) + \theta \sin(\theta) = 0$ Solve this equation for $\theta$.

4. Evaluate $r(\theta)$ at the critical points and the endpoints of the interval $0 \leq \theta \leq \frac{\pi}{2}$.

5. Determine the minimum value of $r(\theta)$ within the interval by comparing the values obtained in the previous step.

6. Round $\theta$ corresponding to the minimum $r(\theta)$ to the nearest thousandth.

Let me perform these calculations for you.The value of $\theta$ in the interval $0 \leq \theta \leq \frac{\pi}{2}$ that corresponds to the point on the curve with the shortest distance from the pole is approximately $\theta = 0.438$ radians. The corresponding minimum value of $r(\theta)$ is approximately $-5.380$.

Would you like more details or have any questions? Here are five related questions you might consider:

1. How does changing the interval affect the value of $\theta$ that minimizes $r(\theta)$?
2. What is the significance of the function $r(\theta)$ in polar coordinates?
3. Can you find the second derivative to confirm whether this point is a minimum?
4. How would the solution change if the function $r(\theta)$ were different?
5. How does the periodic nature of the trigonometric functions affect the results?

Tip: When dealing with trigonometric functions, it's useful to visualize the graph to better understand the behavior of the function over the given interval.