# 01-02 Water flowing into cylindrical tank

**Problem 01**

Water is flowing into a vertical cylindrical tank at the rate of 24 ft^{3}/min. If the radius of the tank is 4 ft, how fast is the surface rising?

**Solution 01**

Volume of water:

$V = \pi r^2h = \pi(4^2)h = 16\pi h$

$V = \pi r^2h = \pi(4^2)h = 16\pi h$

$\dfrac{dV}{dt} = 16\pi \dfrac{dh}{dt}$

$24 = 16\pi \dfrac{dh}{dt}$

$\dfrac{dh}{dt} = 0.477 \,\, \text{ ft/min }$ *answer*

**Problem 02**

Water flows into a vertical cylindrical tank at 12 ft^{3}/min, the surface rises 6 in/min. Find the radius of the tank.

**Solution 02**

Volume of water:

$V = \pi r^2h$

$V = \pi r^2h$

$\dfrac{dV}{dt} = \pi r^2 \dfrac{dh}{dt}$

$12 = \pi r^2 (0.5)$

$r = \sqrt{\dfrac{12}{0.5\pi}\,} = 2.76 \,\, \text{ ft }$ *answer*

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