Efflux problem is my work valid?

[cepheid]

I have completed this problem, and would like to know if my work is correct:

According to Torricelli's theorem, the velocity of a fluid draining from a hole in a tank is v ~= (2gh)^1/2, where h is the depth of water above the hole. Let the hole have an area A₀, and the cylindrical tank have cross-sectional area A_b >> A₀. Derive a formula for the time to drain the tank completely from an initial depth h₀.

My work:

The volume flow rate out of the hole is equal to the rate of change of the volume in the tank:

$$A_0 v = \frac{dV}{dt} = A_b \frac{dh}{dt}$$

$$v = \sqrt{2gh} = \frac{A_b}{A_0} \frac{dh}{dt}$$

Assuming that we start from t₀ = 0, and that the tank is drained after a time T, we can separate variables and integrate:

$$\int_0^T {dt} = T = \frac{A_b}{A_0} \int_{h_0}^0 {\frac{dh}{\sqrt{2gh}}}$$

One thing that bothered me was that I never made use of the information that A_b >> A₀. I thought at first maybe I was supposed to make some approximation somewhere based on that. But then I dug out my first year physics text and saw that Torricelli's theorem was derived from Bernoulli's eqn, and that this information regarding the two areas was used in the derivation. So maybe that's the only reason they gave it to us. Still, is everything else ok?

 

[HallsofIvy]

Of course, you did use the fact that A[sub[0[/sub]<<A_b when you assumed that you could stop the integral at the top of the hole. You are treating that hole the water is coming out of as a point.

 

[cepheid]

Yeah, makes sense to me if the hole is in the side of the container. But Torricelli's thm is valid for the hole in the bottom of the container, which is what it shows in the diagram. How should it be interpreted then?