Mathematics

Problem

An aircraft is at altitude

y

meters and horizontal distance

x

meters from a radar antenna located at the origin. The antenna’s angle of elevation is

\theta = \arctan\!\left(\dfrac{y}{x}\right)

. At a particular instant, the aircraft is 40 m horizontally from the antenna and 30 m above it, moving so that

\dfrac{dx}{dt} = -5\text{ m/s}

(toward the antenna) and

\dfrac{dy}{dt}=6\text{ m/s}

(climbing). Find

\dfrac{d\theta}{dt}

at that instant.