The image shows a problem involving a kite gliding horizontally while its string is unspooled. The key elements of the problem are:

• The kite is at an altitude of 30 meters.
• The kite's horizontal velocity is 3 m/s.
• We are tasked with determining how fast the angle $\theta$ between the kite string and the horizontal decreases when 60 meters of string have been unspooled.

The image also contains parts of the problem that involve:

• (a) Expressing $\theta$ as a function of the kite's horizontal position $x$.
• (b) Calculating the value of $x$ when 60 meters of string is unspooled.
• (c) Finding the rate at which the angle $\theta$ is decreasing, $\frac{d\theta}{dt}$, given that the horizontal velocity is 3 m/s.

Would you like detailed steps and explanations for each part of this problem?

Here are five related questions to extend this:

1. How is the horizontal position $x$ related to the total length of the string?
2. Why is $\theta = \arccos\left(\frac{x}{60}\right)$ the appropriate expression for the angle?
3. What is the significance of using the derivative $\frac{d\theta}{dt}$ in this context?
4. How do changes in the horizontal velocity of the kite affect $\frac{d\theta}{dt}$?
5. Can you apply this concept to determine the rate of change of the string length as the kite moves?

Tip: Always ensure your units are consistent (e.g., seconds, meters) when dealing with rates of change like velocities and angular velocities.