The radius (in centimeters) of an inflating spherical balloon is modeled by r(t) =…

MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

The radius (in centimeters) of an inflating spherical balloon is modeled by

r(t) = \sqrt{3t^{2}+1},

where

t

is the time in seconds after inflation begins. The volume of a sphere is

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

. (a) Derive a simplified formula for the rate of change of volume with respect to time,

\dfrac{dV}{dt}

, expressed solely in terms of

t

. (b) Compute

\dfrac{dV}{dt}

t = 2

seconds and state the exact value including

\pi

. (c) State the units of

\dfrac{dV}{dt}

and briefly explain their physical meaning in this context.