A hot-air balloon is 50 m horizontally from an observer on level ground

MATHEMATICS · Subjective · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

A hot-air balloon is 50 m horizontally from an observer on level ground. The observer’s angle of elevation satisfies

\theta = \arctan\!\left(\dfrac{h}{50}\right)

, where

h

(in meters) is the balloon’s altitude. When

h = 30\text{ m}

, the altitude is increasing at

\dfrac{dh}{dt}=4\text{ m/s}

. Calculate

\dfrac{d\theta}{dt}

at that instant, to three decimal places.