Solving a DE for Water Leaking from a Tank

[ehabmozart]

Homework Statement

Suppose water is leaking from a tank through a circular hole of area Ah at its bottom. When water leaks through a hole, friction and contraction of the stream near the hole reduce the volume of water leaving the tank per second to

c.Ah.(2gh)^.5

, where c (0 < c < 1) is a empirical constant. Determine a differential equation for the height h of water at time t for the cubical tank shown. The radius of the hole is 2 in., and g = 32ft/s2

Homework Equations

The Attempt at a Solution

I am always confused in setting up a DE. Over here, the answer says that V= Aw(area of water). h . The next step written was that dh/dt = 1/Aw * dV/dt ... How does this make sense.. I am confused here. I understand that the "c.Ah.(2gh)^.5" given back in the question is dv/dt . But when you multiply it with 1/Aw, how do you get dh/dt ? Thanks for any help!

 

[Ray Vickson]

Homework Statement

Suppose water is leaking from a tank through a circular hole of area Ah at its bottom. When water leaks through a hole, friction and contraction of the stream near the hole reduce the volume of water leaving the tank per second to

c.Ah.(2gh)^.5

, where c (0 < c < 1) is a empirical constant. Determine a differential equation for the height h of water at time t for the cubical tank shown. The radius of the hole is 2 in., and g = 32ft/s2

Homework Equations

The Attempt at a Solution

I am always confused in setting up a DE. Over here, the answer says that V= Aw(area of water). h . The next step written was that dh/dt = 1/Aw * dV/dt ... How does this make sense.. I am confused here. I understand that the "c.Ah.(2gh)^.5" given back in the question is dv/dt . But when you multiply it with 1/Aw, how do you get dh/dt ? Thanks for any help!

The area of the hole is Ah? How can the hole's area depend on the height of water in the tank?

 

[Mark44]

The area of the hole is Ah? How can the hole's area depend on the height of water in the tank?

I think the OP means this as A_h, not A * h. Also, I think Aw means A_w.

ehabmozart,
You can make what you write clearer by using the features available on this site. For example, to write exponents and subscripts, click the Go Advanced button below the input area, which causes the advanced menu to open across the top. One button is X², which you can use to write exponents. Another button is X₂, which you can use to write subscripts.

 

[SteamKing]

I am always confused in setting up a DE. Over here, the answer says that V= Aw(area of water). h . The next step written was that dh/dt = 1/Aw * dV/dt ... How does this make sense.. I am confused here. I understand that the "c.Ah.(2gh)^.5" given back in the question is dv/dt . But when you multiply it with 1/Aw, how do you get dh/dt ? Thanks for any help!

If you take the equation V = Aw * h and differentiate with respect to 't', what do you get?