Question

A hot air balloon, 50 feet from an observer, is rising at 20 ft/sec. At 5 seconds after lift off
1. How fast is the distance between the observer and the balloon changing?

2. How fast is the angle of elevation changing?

Answer 1

the distance z at height x is

z^2 = 50^2 + x^2
so,
2z dz/dt = 2x dx/dt

find z when x = 5*20=100, and plug and chug.

tanθ = x/50
sec^2 θ dθ/dt = 1/50 dx/dt