What is the rate of change of the cameras angle?

Question

A rocket is launched so that it rises vertically. A camera is positioned 15000 ft from the launch pad. When the rocket is 6000 ft above the launch pad, its velocity is 300 ft/s. Find the necessary rate of change of the camera's angle as a function of time so that it stays focused on the rocket. Leave your answer as an exact number.

Answer 1

Let Θ: angle of inclination of the camera (in radians), h: vertical height of the rocket (in ft)

dh/dt = 300 ft/sec ; instant: h = 6,000 so secΘ = 5 / √29 ; dΘ/dt = ?

tanΘ = h / 15,000

sec²Θ · dΘ/dt = 1/15,000 · dh/dt

25/29 · dΘ/dt = 1 / 50

dΘ/dt = 29 / 1,250 rads/sec