During a laboratory test

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

Problem

During a laboratory test, a spherical probe is inflated so that its volume increases at a constant rate of

150\text{ cm}^3\!/\text{s}

starting from a negligible initial size at

t=0

. Let

r(t)

be the radius (in centimeters) at time

t

seconds, and let

S(t)=4\pi r^2

denote its surface area. (a) Write an equation that relates

\dfrac{dr}{dt}

to the given constant volumetric rate. (b) Determine

\dfrac{dr}{dt}

at the instant when

r=10\text{ cm}

. (c) Obtain an expression for

\dfrac{dS}{dt}

in terms of

r

and

\dfrac{dr}{dt}

. Then evaluate

\dfrac{dS}{dt}

when

r=10\text{ cm}

and state whether the surface area is increasing or decreasing at that instant. (d) Find all positive radii

r

for which

\dfrac{dS}{dt}=0

. Assuming the sphere started from radius

0

, determine the corresponding time(s) and explain what happens to the surface area before and after that time.