1. ## Calculus Word Problem

Water is leaking out of an inverted conical tank at a rate of 7000 cm3/min at the same time that water is being pumped into the tank at a constant rate. The tank has height 6 m and the diameter at the top is 4 m. If the water level is rising at a rate of 20 cm/min when the height of the water is 2 m, find the rate at which water is being pumped into the tank.

2. Originally Posted by kashifzaidi
Water is leaking out of an inverted conical tank at a rate of 7000 cm3/min at the same time that water is being pumped into the tank at a constant rate. The tank has height 6 m and the diameter at the top is 4 m. If the water level is rising at a rate of 20 cm/min when the height of the water is 2 m, find the rate at which water is being pumped into the tank.

$\displaystyle \frac{r}{h} = \frac{2}{6}$

$\displaystyle r = \frac{h}{3}$

$\displaystyle V = \frac{\pi}{3}\left(\frac{h}{3}\right)^2 h$

$\displaystyle V = \frac{\pi}{27} \cdot h^3$

$\displaystyle \frac{dV}{dt} = \frac{\pi}{9} \cdot h^2 \cdot \frac{dh}{dt}$ = (volume rate in) - (volume rate leaking out)