Air is pumped into a spherical balloon at a rate of (liters per second)

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

Problem

Air is pumped into a spherical balloon at a rate of

5\text{ L/s}

(liters per second).

1\text{ L}=1000\text{ cm}^3

. The volume

V

(in

\text{cm}^3

) of a sphere with radius

r

centimeters is

V = \dfrac{4}{3}\pi r^{3}

. At the instant when the radius is

15\text{ cm}

, the balloon has room to expand freely. (a) Calculate the instantaneous rate of change of the radius at this instant. Express your answer in

\text{cm/s}

. (b) Using the value of

\dfrac{dr}{dt}

found in part (a) as an approximation over a short time interval, estimate the radius of the balloon

10

seconds later. Clearly state any assumptions made in your estimation.