A weather observer stands on level ground 200 m east of a rocket launch site

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

Problem

A weather observer stands on level ground 200 m east of a rocket launch site. At time

t=0

, the rocket lifts off vertically from the launch pad. Let

y(t)

be the height (in meters) of the rocket above the ground at time

t

seconds, and let

\theta(t)

be the angle (in radians) between the observer’s line of sight to the rocket and the horizontal ground. At the instant when the rocket is 600 m above the ground, its upward velocity is 80 m/s and its upward acceleration is 12 m/s

^2

. (a) Derive an expression for

\dfrac{d\theta}{dt}

in terms of

y

\dfrac{dy}{dt}

, and the fixed horizontal distance between the observer and the launch site. Then evaluate

\dfrac{d\theta}{dt}

(including its sign) at the instant described. (b) Using your work from part (a), determine the instantaneous rate of change of

\dfrac{d\theta}{dt}

with respect to time (that is, compute

\dfrac{d^2\theta}{dt^2}

) at the same instant. State whether the observer’s angle of elevation is increasing at an increasing rate, increasing at a decreasing rate, or decreasing, and justify your conclusion.

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