A soap bubble is expanding so that its radius (in centimeters) changes with time…

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

Problem

A soap bubble is expanding so that its radius

r(t)

(in centimeters) changes with time according to the model

\displaystyle \frac{dr}{dt}=0.015t+0.010, \quad t\ge 0,

where

t

is measured in seconds. Let

V(t)=\dfrac{4}{3}\pi r(t)^3

denote the volume of the bubble and

S(t)=4\pi r(t)^2

its surface area. Define the efficiency ratio of the bubble by

E(t)=\dfrac{V(t)}{S(t)}.

Answer the following questions. (a) Obtain an explicit expression for

\dfrac{dE}{dt}

in terms of

r

and

\dfrac{dr}{dt}

. Simplify your result completely. (b) At

t=4\text{ s}

the radius is

r=1.16\text{ cm}

. Compute

\dfrac{dE}{dt}

at this instant and state the units of your answer. (c) Determine the exact time

t>0

at which the efficiency ratio

E(t)

attains its maximum value. Justify that the value you obtain indeed corresponds to a maximum. (d) Physically interpret what it means for

\dfrac{dE}{dt}

to be zero for the expanding bubble.