An observer stands on a straight shoreline and watches a boat travel on a river that…

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

Problem

An observer stands on a straight shoreline and watches a boat travel on a river that runs straight and horizontal relative to the observer’s view. The observer is 60 m north of a point

A

on the shore. The boat moves eastward along a line that is 40 m south of

A

at a speed of 5 m/s. Let

\theta(t)

be the angle (in radians) between the observer’s line of sight to the boat and the due-east shoreline, measured positive clockwise from due east. The straight-line distance from the observer to the boat at time

t

seconds is

r(t)

meters. (a) Express

\tan \theta

in terms of

r

and known constants by introducing a convenient coordinate system. (b) Eliminate

r

to write

\tan \theta

solely as a function of the boat’s eastward distance

x(t)

from point

A

. (c) Differentiate the expression from part (b) with respect to time to obtain

\dfrac{d\theta}{dt}

in terms of

x

and

\dfrac{dx}{dt}

. (d) Determine

\dfrac{d\theta}{dt}

, in rad/s, at the moment the boat is 120 m east of point

A

. State whether the absolute value of the angular speed is increasing or decreasing as the boat continues eastward, and justify your conclusion using your expression from part (c).

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