An observer stands on level ground 120 ,m east of the point where a hot–air balloon…

MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

An observer stands on level ground 120\,m east of the point where a hot–air balloon is lifting off vertically. The observer measures the balloon’s angle of elevation

\theta(t)

(in radians) above the horizontal at time

t

seconds. At the instant when the balloon is 80\,m above the ground, its vertical velocity is 6\,m\,s

^{-1}

. (a) Write an equation that relates

\theta

and the balloon’s height

h

at any time. (b) Differentiate your equation with respect to time to obtain an expression for

\dfrac{d\theta}{dt}

in terms of

h

and

\dfrac{dh}{dt}

. (c) Calculate

\dfrac{d\theta}{dt}

, in radians per second, at the instant when

h=80

\,m. (d) Using appropriate geometric or analytic reasoning, determine the rate at which the straight–line distance from the observer to the balloon is changing at that same instant. Include units in your answer.

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