1. ## Calc word problem

Sand is draining from a conical filter into a cylindrical pot at a rate of 10 cubic centimeters per minute. The conical filter measures 15 cm tall and 15 cm across its top. The diameter of the pot is 15 cm.

(a) How fast is the level of sand in the filter falling when the coffee in the filter is 8 cm deep?

(b) At the same time and under the same conditions as in (a), how fast is the level of the sand in the pot rising?

Sand is draining from a conical filter into a cylindrical pot at a rate of 10 cubic centimeters per minute. The conical filter measures 15 cm tall and 15 cm across its top. The diameter of the pot is 15 cm.

(a) How fast is the level of sand in the filter falling when the coffee in the filter is 8 cm deep?

(b) At the same time and under the same conditions as in (a), how fast is the level of the sand in the pot rising?
part (a) ...

$\frac{r}{h} = \frac{1}{2}$

$r = \frac{h}{2}$

$V = \frac{\pi}{3}r^2 h$

$V = \frac{\pi}{12}h^3$

take the time derivative of the above equation ... you know that $\frac{dV}{dt} = -10 \, cm^3/min$ and $h = 8 \, cm$ ... solve for $\frac{dh}{dt}$

part (b) ...

$\frac{dV}{dt} = +10 \, cm^3/min$

$V = \pi r^2 h$

$r$ is a constant ...

$V = \pi (7.5)^2 h$

take the time derivative and solve for $\frac{dh}{dt}$