The illuminance (in lux) from a point light source of luminous power (in lumens) at a…

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

Problem

The illuminance

I

(in lux) from a point light source of luminous power

P

(in lumens) at a distance

r

(in meters) obeys the inverse-square relationship

I=\dfrac{P}{4\pi r^{2}}.

A battery-powered drone carries a light whose power output is decreasing as the battery drains, while the drone simultaneously moves away from a sensor that measures the illuminance. At an instant when

P=1.20\times10^{3}\text{ lm}

and

r=12.0\text{ m}

, the power is decreasing at

\dfrac{dP}{dt}=-18.0\text{ lm/s}

and the distance is increasing at

\dfrac{dr}{dt}=2.40\text{ m/s}

. (a) Derive an expression for the rate of change

\dfrac{dI}{dt}

in terms of

P,\,\dfrac{dP}{dt},\,r

, and

\dfrac{dr}{dt}

. (b) Evaluate

\dfrac{dI}{dt}

at the given instant and state the units. (c) Determine the percent rate of change of illuminance per second at that instant and interpret the result in the context of the sensor’s observations.

Related theory

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