A research drone is flying so that its horizontal position (in meters east of a fixed…

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

A research drone is flying so that its horizontal position (in meters east of a fixed origin) and altitude (in meters above ground) at time

t

seconds are given by

x(t) = 120 + 40t, \quad y(t) = 50 + 30t.

At time

t=0

, a cable of length 130 m is attached between the drone and a ground station located at the origin

(0,0)

. As the drone moves, the length

L(t)

of the cable changes accordingly. (a) Derive a formula for

\dfrac{dL}{dt}

in terms of

x

y

\dfrac{dx}{dt}

, and

\dfrac{dy}{dt}

. Do not substitute numerical values in this part. (b) Compute

\dfrac{dL}{dt}

at time

t = 2

s, and interpret the sign and magnitude of your result in the physical context of the problem. (c) Determine the earliest time

t>0

(if any) at which the cable begins to shorten rather than lengthen. Justify your answer.