# Rate of Change of Volume of Sand in Conical Shape

**Problem**

A conveyor is dispersing sands which forms into a conical pile whose height is approximately 4/3 of its base radius. Determine how fast the volume of the conical sand is changing when the radius of the base is 3 feet, if the rate of change of the radius is 3 inches per minute.

A. 2π ft/min | C. 3π ft/min |

B. 4π ft/min | D. 5π ft/min |

**Answer Key**

[ C ]

**Solution**

$h = \dfrac{4}{3}r$

$\dfrac{dh}{dt} = \dfrac{4}{3} \cdot \dfrac{dr}{dt}$

$\dfrac{dh}{dt} = \dfrac{4}{3} \cdot \dfrac{3}{12}$

$\dfrac{dh}{dt} = \dfrac{1}{3} ~ \text{ft/min}$

$Q = vA$

$\dfrac{dV}{dt} = \dfrac{1}{3} \cdot \pi(3^2)$

$\dfrac{dV}{dt} = 3\pi ~\text{ft/min}$ ← *answer*

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