A hot–air balloon ascends vertically at a constant and remains directly above a point…

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

Problem

A hot–air balloon ascends vertically at a constant

2.0\text{ m/s}

and remains directly above a point on level ground located

100\text{ m}

from a spotlight. The spotlight swivels so that its beam always points at the balloon. Let

\theta(t)

be the angle (in radians) between the beam and the ground. (a) Express

\theta

as a function of the balloon’s height

h

above the ground. (b) Compute the angular speed

\dfrac{d\theta}{dt}

of the spotlight at the instant when the balloon is

50\text{ m}

above the ground. Give units with your answer. (c) Estimate the balloon’s height when the angular speed of the spotlight equals

0.010\text{ rad/s}

. Indicate any approximations made in your reasoning. (d) Describe how

\dfrac{d\theta}{dt}

behaves as the balloon continues to rise indefinitely, and interpret this behavior in terms of the spotlight operator’s experience.