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Subject: "related rates with cone lying on side instead of apex down."     Previous Topic | Next Topic
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Conferences The CTK Exchange College math Topic #655
Reading Topic #655
megatron1
Member since Oct-31-07
Oct-31-07, 06:14 PM (EST)
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"related rates with cone lying on side instead of apex down."
 
   I have an interesting and challenging take on the old related rates problem involving a cone being filled with water.

Suppose a cone with radius 10 feet and height 24 feet is being filled at the rate of 20 ft^3/min. How fast is the water rising when the height of the water is 6 feet?." Only instead of the cone being apex down (which is easy using similar triangles), it is lying on its side.

I know when you slice a cone with a plane you get a parabola. Therefore, the 'slices' will be a parabola sliced by a horizontal plane. The trick is getting this into an equation I can differentiate implcitly. Using the plane z=ay+c and the cone
z^2=a^2(x^2+y^z), I managed to derive y=ax^2/(2c)+c/(2a).

Now, what?. Does anyone have any insight?. I really would like to see this through. By the way, this is an interesting problem posed by a friend, not homework.

Thanks for any help or advice.

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alexbadmin
Charter Member
2122 posts
Nov-01-07, 11:15 AM (EST)
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1. "RE: cone lying on side"
In response to message #0
 
   Just a few thoughts:

Pass to the areas A(c) of the parabolas. Express those areas in terms of c. The volume V(c) then will be the integral of those areas. You are interested in a function c = c(t) for which dV/dt is constant. But dV/dt = dv/dc · dc/dt, so that A(c) · c'(t) is constant. Related c to the height of water.


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