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#1 2005-10-14 14:23:44

Saitenji
Member
Registered: 2005-09-24
Posts: 10

Depth in Calculus

A conical tank (with vertext down) is 10 ft across the top and 12 feet deep. If water is flowing into the tank at a rate of 10ft³/min, Find the rate of change of the depth of the water when the water is 8ft deep.

...I'm pretty sure it's asking for dh/dt (or dD/dt, OR Rate of change of the depth.) Problem is...I can't seem to figure out what formula to use...
Any help would be very much appreciated. Thanks in advance!

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#2 2005-10-14 16:28:29

ganesh
Moderator
Registered: 2005-06-28
Posts: 13,593

Re: Depth in Calculus

Volume of the concial tank is 1/3*pi*r²*h
We know the diameter is 10 feet, therefore, radius is 5 feet.
The height is 12 feet, therefore radius is 5/12*h
V = 1/3*pi*(5/12 *h)²*h
V = 1/3*pi*5/12*h³
V = 5/36*pi*h³
dV/dt = 5/36*pi*3h² dh/dt
We know dV/dt=10 ft³/min
Therefore, when h=8,
10 = 5/12*pi*64*dh/dt
dh/dt = (10*12)/(5*pi*64)
dh/dt= 120/320*pi
dh/dt = 0.1194 ft
I haven't checked for calculation errors!


Character is who you are when no one is looking.

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#3 2005-10-14 17:43:10

ryos
Member
Registered: 2005-08-04
Posts: 394

Re: Depth in Calculus

The formula for the volume of a cone is V = (1/3) * (pi * r²) * h.

The volume of water in the tank is a function of time:
V(t) = 10 (ft³ / min) * t (min).

As the water flows into the tank, it fills a smaller cone in the tip of the larger one, the volume of which is the volume of water that has entered the tank so far. The radius also varies with time, and this will give us trouble unless we do something about it now.

You can think of a cone as a right triangle that has been spun about its height. This triangle will let us find r as a function of h, so we can eliminate r altogether in our original formula.

For the cone given in this problem, r = h * tan[sin-¹(5/12)]. You should probably draw the triangle to convince yourself that this is true.

Ok, now plug that in for radius in our original formula:
V = (1/3) pi * h * {h * tan[sin-¹(5/12)] }²

We need h as a function of time, so start by solving for h:
h = (3V / (pi * tan[sin-¹(5/12)]² ))^(1/3)
Oof.

Remember that V is a function of time, so we can substitute that to get h as a function of time:
h(t) = (30t / (pi * tan[sin-¹(5/12)]² ))^(1/3)

Differentiate this function you must. I don't want to wink.

Too bad they didn't just give you a time. Instead, you have to find the rate of change of h when h = 8. You can find the time at which this happens by setting h = 8 and solving for t. Plug that t into h'(t), and you should be set.

Last edited by ryos (2005-10-14 17:46:35)


El que pega primero pega dos veces.

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#4 2005-10-14 17:55:53

ryos
Member
Registered: 2005-08-04
Posts: 394

Re: Depth in Calculus

Heh, I took too long, ganesh beat me to it.

I also didn't check for calculation errors. And ganesh's expression for r is a lot simpler than mine. Did I mention that I drag at trig? It's pretty much my least favorite part of math.

He's right, of course. tan(some angle) = O / A, so tan[sin-¹(5/12)]h = (5/12)h. Neat!


El que pega primero pega dos veces.

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#5 2005-10-14 18:02:20

ryos
Member
Registered: 2005-08-04
Posts: 394

Re: Depth in Calculus

Hey ganesh, I was trying to learn from you wink, and I found a calculation error:

V = 1/3*pi*(5/12 *h)²*h
V = 1/3*pi*5/12*h³

You forgot to square the 5/12.

Also, dh/dt should have units of ft/min. I have no idea what, if anything, went wrong.


El que pega primero pega dos veces.

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#6 2005-10-14 18:28:15

ganesh
Moderator
Registered: 2005-06-28
Posts: 13,593

Re: Depth in Calculus

OMG...
I forgot to square 5/12.
Thats right, I hadn't checked for calculation erros.
Heisenberg's Uncertainity Principle:-
You cannot work at your 100% speed with 100% accuracy smile


Character is who you are when no one is looking.

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#7 2005-10-14 18:41:58

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,535

Re: Depth in Calculus

You'll never catch up with my record of mistakes!


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#8 2005-10-14 21:28:30

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Depth in Calculus

Such as not giving insomnia any boards to mod?
Or was that intentional, for some odd reason?


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

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#9 2005-10-14 23:32:34

justlookingforthemoment
Moderator
Registered: 2005-05-26
Posts: 2,161

Re: Depth in Calculus

Well...insomnia said he wanted to be a 'mod'. He never said he wanted to mod.

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