A drone equipped with a downward-facing spotlight flies horizontally at a constant…

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

Problem

A drone equipped with a downward-facing spotlight flies horizontally at a constant altitude of

15\,\text{m}

above a straight jogging path. At time

t = 0

, the drone is directly above a runner who is

1.8\,\text{m}

tall and standing still on the path. The drone then moves horizontally along the path with constant speed

5.0\,\text{m/s}

. Let •

x(t)

be the horizontal distance (in meters) between the drone and the runner, •

y(t)

be the length (in meters) of the runner’s shadow on the ground cast by the spotlight. (a) Obtain an explicit equation relating

x

and

y

at any time

t

. (b) Differentiate the equation from part (a) with respect to time to express

\frac{dy}{dt}

in terms of

x

\frac{dx}{dt}

, and any required constants. (c) When the drone is

20\,\text{m}

horizontally away from the runner, determine (i) the instantaneous rate of change of the shadow length,

\frac{dy}{dt}

, and (ii) the speed of the tip of the runner’s shadow along the ground. Give numerical answers with appropriate units.