A weather balloon is tethered to a stake on a level field 100 ,m from a ground observer

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

Problem

A weather balloon is tethered to a stake on a level field 100\,m from a ground observer. The observer measures the angle of elevation

\theta

(in radians) to the balloon. At a particular instant the balloon is directly above a point on the ground 100\,m from the observer, and the balloon’s angle of elevation is

\theta_0=\dfrac{\pi}{3}

. At that instant the angle of elevation is increasing at a rate of

\dfrac{d\theta}{dt}=0.01\,\text{rad}\,\text{s}^{-1}

. The balloon moves only vertically. (1) Express the balloon’s height

h

(in metres) above ground level as a function of

\theta

. Clearly define any additional variables you introduce. (2) Determine the instantaneous rate of change

\dfrac{dh}{dt}

of the balloon’s height at the instant described. Include correct units in your answer. (3) Interpret the sign and magnitude of

\dfrac{dh}{dt}

in the context of the situation.

Related theory

Browse the problem library