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nicholaskeyser

Guest

Help study for Calc II final.

I need this quickly to study for a final tommorrow

8.    Let h(t) and V(t) represent the height and volume of water in a tank at time t. If water leaks through a hole with area a at the bottom of the tank, then Torricelli’s Law says that  dV/dt =  -a (2gh)^(1/2), where g is the acceleration due to gravity. So, the rate at which water flows from the tank is proportional to the square root of the water height.

a.    Suppose the tank is cylindrical with height 6 feet and radius 2 feet. The hole is circular with radius 1 inch. If g = 32 feet per second squared, show that 
.a.)  dh/dt = (-1/72) (h)^(1/2)

b.    Assuming that the tank is full at time t = 0, show that   is a solution to the differential equation in (a)
h= ((-1/144)*t + 6^(1/2)) ^2

c.    How long will it take for the water in the tank to drain completely?

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Help study for Calc II final.

For a), replace the dV in Torricelli's Law with dh, by using V = πr²h.
Then use the measurements given to replace a and g with numbers, and it should simplify to the given result.

For b), you need to separate the variables because there's an h on the right-hand side.
Rearrange it to get (1/√h)dh = -1/72dt, then integrate each side with respect to the appropriate variable.

This will give you that h = [something] + C.
Use the fact that the cylinder is full initially to get some information that will let you find C.

For c), simply work out what value of t will give h=0 in the equation you just found for b).

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Why did the vector cross the road?
It wanted to be normal.

cregstephen

Member

Registered: 2008-12-30

Posts: 6

Re: Help study for Calc II final.

Pls help me with fundamental theorems in calculus  like..

The second fundamental theorem of calculus states that
F(x) = ò a x f(t) dt

then F '(x) = f(x).