A solid sphere of variable radius r(t) has mass m that remains constant in time while…
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A solid sphere of variable radius r(t) has mass m that remains constant in time while its density ρ(t) changes because of internal compression. The sphere’s mass is given by m=34πr3ρ. Both r(t) and ρ(t) are differentiable functions of time t.
(a) Differentiate the mass equation implicitly with respect to time to obtain a relationship between dtdr and dtdρ. Do not solve for either derivative.
(b) Describe in words what your result in part (a) indicates about how the sphere’s radius must change whenever its density is increasing.