Hot air balloon rising

Question

Madeleine watches a hot air balloon launch from 400 meters away. If the balloon is rising vertically at a rate of 7 meters/second, at what rate is the angle of elevation changing from Madeleine's perspective when the balloon is 700 meters in the air? Answer (in radians per second):

Answer 1

Imagine a right triangle with a base of x=400 meters and a height of y. dy/dt = 7 m/s

tan(θ) = y/400

Take the derivative of both sides, with respect to t:

sec²(θ) dθ/dt = (1/400) dy/dt

dθ/dt = (1/sec²(θ))(1/400) dy/dt = (cos²(θ)/400)(7)

When y = 700, hypotenuse = √(400²+700²)

cos²(θ)=160000/(400²+700²) = 160000/650000 = 16/65

dθ/dt = (16/65)(7/400) = 0.0043 rad/s