A drone transmits a radio signal of constant power

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

Problem

P=100\text{ W}

. The signal intensity

S(t)

(in watts per square meter) received at a ground station is modeled by

S(t)=\dfrac{P}{4\pi d(t)^2},

where

d(t)

meters is the time-dependent straight-line distance between the drone and the ground station. At a certain moment the drone is

50\text{ m}

away and is approaching the station at a horizontal speed of

2\text{ m s}^{-1}

; that is,

\dfrac{dd}{dt}=-2\text{ m s}^{-1}

at that instant. Answer the questions below. (a) Obtain a formula for

\dfrac{dS}{dt}

in terms of

P

d

, and

\dfrac{dd}{dt}

. Simplify as much as possible. (b) Compute

\dfrac{dS}{dt}

at the instant described above. Provide the correct units with your answer. (c) Determine the distance

d

(in meters) at which the magnitude of

\dfrac{dS}{dt}

equals

0.08\text{ W m}^{-2}\text{ s}^{-1}

. Assume

\dfrac{dd}{dt}

remains

-2\text{ m s}^{-1}

while the drone approaches.