Sand is poured onto the ground forming a right circular cone

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

Problem

Sand is poured onto the ground forming a right circular cone. At a certain moment the pile has radius

15\text{ cm}

and height

10\text{ cm}

. The radius is increasing at

\dfrac{dr}{dt}=\dfrac13\text{ cm/s}

, and sand is being added at

\dfrac{dV}{dt}=20\text{ cm}^3\!\big/\!\text{s}

. Given the cone volume formula

V=\tfrac13 \pi r^{2}h

, which of the following is the value of

\dfrac{dh}{dt}

at that instant? (a)

\displaystyle \frac{60-100\pi}{75\pi}\text{ cm/s}

(b)

\displaystyle \frac{100\pi-60}{225\pi}\text{ cm/s}

(c)

\displaystyle \frac{60-100\pi}{225\pi}\text{ cm/s}

(d)

\displaystyle \frac{20-100\pi}{225\pi}\text{ cm/s}