Car A travels north along a straight road and car B travels east along another…

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

Problem

Car A travels north along a straight road and car B travels east along another straight road that meets the first at right angles at point

O

. At noon, car A is

12

km south of

O

and moving northward at

60

km h

^{-1}

, while car B is

9

km west of

O

and moving eastward at

80

km h

^{-1}

. (i) Let

x(t)

km be the northward distance of car A from

O

and

y(t)

km the eastward distance of car B from

O

at time

t

hours after noon. Write an expression for the square of the distance

D(t)

km between the two cars in terms of

x

and

y

. (ii) Derive a formula for

\dfrac{dD}{dt}

in terms of

x

y

\dfrac{dx}{dt}

and

\dfrac{dy}{dt}

. (iii) Calculate

\dfrac{dD}{dt}

at 12:15 p.m. (iv) State, with reason, whether the distance between the cars is increasing or decreasing at that moment.