A circular parachute canopy is being folded so that its radius decreases at a…

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

Problem

A circular parachute canopy is being folded so that its radius

r

decreases at a constant rate of

0.05\text{ m\,s}^{-1}

at a particular instant. Let

A

denote the area of the canopy at that instant. (a) Without performing an explicit differentiation, explain qualitatively how the rate of change

\dfrac{dA}{dt}

depends on the instantaneous radius

r

and on the sign of

\dfrac{dr}{dt}

. State whether

\dfrac{dA}{dt}

is positive, negative, or zero at the instant described and justify your answer. (b) The same canopy was previously observed when its radius was

2\text{ m}

and was shrinking at the *same* constant speed

0.05\text{ m\,s}^{-1}

. Compare the *magnitude* of

\dfrac{dA}{dt}

r=4\text{ m}

and at

r=2\text{ m}

. Explain which radius corresponds to the faster loss of area and why. (c) Identify the correct units for

\dfrac{dA}{dt}

in this context and interpret the meaning of a negative value of

\dfrac{dA}{dt}

for the parachute canopy.