Mathematics

Mathematics

Problem

A spherical balloon contains helium whose mass is M(t)M(t) kilograms, while its volume is V(t)=43πr(t)3V(t)=\dfrac{4}{3}\pi r(t)^3 cubic meters. At an instant when the radius is 1.0 m1.0\text{ m} and the mass is 0.75 kg0.75\text{ kg}, the radius is increasing at drdt=0.05 m s−1\dfrac{dr}{dt}=0.05\text{ m s}^{-1} and helium is leaking out at dMdt=−0.02 kg s−1\dfrac{dM}{dt}=-0.02\text{ kg s}^{-1}. The density of helium inside the balloon is ρ(t)=M(t)V(t)\rho(t)=\dfrac{M(t)}{V(t)}. What is dρdt\dfrac{d\rho}{dt} at that instant (in kg m−3 s−1\text{kg m}^{-3}\text{ s}^{-1})? a) Approximately 0.0320.032 b) Approximately −0.008-0.008 c) Approximately −0.032-0.032 d) Approximately 0.0080.008

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