The volume of a conical pile of sand is increasing at a rate of $576 \pi$ ft³/min, and the height of the pile always equals the radius r of the base. Express r as a function of time t (in minutes), assuming that r = 0 when t = 0. A) $r = 12 \sqrt [ 3 ] { t }$ B) $r = \sqrt [ 3 ] { 576 \pi t }$ C) $r = \sqrt [ 3 ] { 12 t }$ D) $r = 576 \pi \sqrt [ 3 ] { t }$ E) $r = 576 \sqrt [ 3 ] { \pi t }$

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Quizplus · Accepted Answer

The volume of a cone is $V = \frac{1}{3} \pi r^2 h$. Given $h = r$, $V = \frac{1}{3} \pi r^3$. Differentiating with respect to time, $\frac{dV}{dt} = \pi r^2 \frac{dr}{dt}$. Given $\frac{dV}{dt} = 576\pi$, solve for $\frac{dr}{dt}$ and integrate to find $r$ as a function of $t$.

The Volume of a Conical Pile of Sand Is Increasing 576π576 \pi576π

The Volume of a Conical Pile of Sand Is Increasing $576 \pi$