Need a bit of Mathematics C help any nerds out there

Imagine this, a cylindrical tank with a liquid level height ‘h’. The tank develops a leak in the bottom; Torricelli’s Law says that “Water in an open tank will flow out through a small hole in the bottom with the velocity it would acquire in falling freely from the water level, h.”

If a mass of water ‘m’ fell freely from height ‘h’, the GPE of the mass would be transformed into KE.

*mgh=**1**2**m**v**2* [IMG]file:///C:\DOCUME~1\Operator\LOCALS~1\Temp\msohtmlclip1\01 \clip_image002.gif[/IMG]

*2gh=**v**2* [IMG]file:///C:\DOCUME~1\Operator\LOCALS~1\Temp\msohtmlclip1\01 \clip_image004.gif[/IMG]

Therefore according to Torricelli’s Law, the velocity that the water exits the tank is *v=**2gh*, where g = 9.8m/s^{2}.

The change in volume of the tank *dV**dt**=**(V**entering**s)/s**-**(V(leaving))/**s*

- Find a relationship for the amount of water leaving the tank per unit time (
*dV**dt* [IMG]file:///C:\DOCUME~1\Operator\LOCALS~1\Temp\msohtmlclip1\01 \clip_image002.gif[/IMG], m^{3} per second) using the area of the hole and the velocity of the water. (Volume entering is 0 L/s)

- Use the chain rule
*dV**dt**=**dh**dt**×**dV**dh *to develop the differential equation *dh**dt**=f(h)* . Solve the equation to find the relationship *h(t)* [IMG]file:///C:\DOCUME~1\Operator\LOCALS~1\Temp\msohtmlclip1\01 \clip_image008.gif[/IMG].

A full water tank is 1.8m high and has a radius of 2m, a small hole of area 5cm^{2} is created in the bottom of the tank, how long does it take for the tank to empty.

- Consider a tank of the same dimensions as above. The tank is half full when the hole develops and water now enters the tank at a rate of 10L/s. How long does it take to fill/empty the tank?

[IMG]file:///C:\DOCUME~1\Operator\LOCALS~1\Temp\msohtmlclip1\01 \clip_image008.gif[/IMG]