A ground-based radar antenna is located at the origin of a coordinate system

MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives

Problem

A ground-based radar antenna is located at the origin of a coordinate system. A plane flies directly away along the positive

x

-axis at a constant altitude of 3 km. At time

t

seconds the plane’s horizontal distance from the radar is

x(t)

kilometers, and the radar sweeps upward through an angle

\theta(t)

radians measured from the horizontal. (a) Write an equation that relates

\tan \theta

x

using the given altitude. (b) Differentiate your equation with respect to time

t

to obtain a relationship between

\dfrac{d\theta}{dt}

and

\dfrac{dx}{dt}

. (c) The plane’s ground speed is 250 m/s. Determine

\dfrac{d\theta}{dt}

, in rad/s, at the instant when

x=4

km. (d) Explain how

\dfrac{d\theta}{dt}

would change if the plane were climbing at a small constant rate instead of maintaining a fixed altitude, assuming the horizontal speed stays the same. No calculation is required—support your answer with reasoning based on your result in part (b).

Related theory

Browse the problem library