A spherical balloon contains helium whose mass is kilograms

MATHEMATICS · Single correct · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

A spherical balloon contains helium whose mass is

M(t)

kilograms, while its volume is

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

cubic meters. At an instant when the radius is

1.0\text{ m}

and the mass is

0.75\text{ kg}

, the radius is increasing at

\dfrac{dr}{dt}=0.05\text{ m s}^{-1}

and helium is leaking out at

\dfrac{dM}{dt}=-0.02\text{ kg s}^{-1}

. The density of helium inside the balloon is

\rho(t)=\dfrac{M(t)}{V(t)}

. What is

\dfrac{d\rho}{dt}

at that instant (in

\text{kg m}^{-3}\text{ s}^{-1}

)? (a) Approximately

0.032

(b) Approximately

-0.008

-0.032

(d) Approximately

0.008