An ice sphere melts so that its radius (in centimeters) decreases at the constant rate

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

Problem

An ice sphere melts so that its radius

r(t)

(in centimeters) decreases at the constant rate

\dfrac{dr}{dt}=-0.05\,\text{cm/min}

. The surface-area–to–volume ratio is

S(t)=\dfrac{4\pi r(t)^{2}}{\tfrac{4}{3}\pi r(t)^{3}}=\dfrac{3}{r(t)}

. (a) Derive a formula for

\dfrac{dS}{dt}

in terms of

r(t)

and

\dfrac{dr}{dt}

. (b) The melting begins with radius

r(0)=5\,\text{cm}

. Determine the time, in minutes, at which

\dfrac{dS}{dt}=0.03\,\text{cm}^{-2}\text{/min}