Pump out the water from a cone?

mrjust

mrjust

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Dec 12, 2012
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I'm in my first semester of calculus. Heres the question I don't understand. I know I have to use W = F * D.

A cone with height 12 ft and radius 4 ft, pointing down-
ward, is filled with water to a depth of 9 ft. Find the work
required to pump all the water out over the top.

Where x is the radius of the slab and y is the height of the cone.

1.) I cut out a circular slab from the cone with radius [ x= (12-y)/3 ]
2.) Now I have [ pi( ( 12-y )/3 )^2(delta(y))
3.) Density of water is 62.4
4.) Volume = [ pi( ( 12-y )/3 )^2(delta(y)) ]62.4
5.) I set up the integral from 0 to 9 of [ pi( ( 12-y )/3 )^2]62.4*y dy ]

I keep on getting the wrong answer. Answer in the book is 27,788 ft-lbs.
I don't understand what I'm doing wrong.

Edit:
Fixed Integral function
 
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You did not define your x and y, but it appears to me that you set up the integral wrongly; I see both delta-x and dy.


If you introduced a coordinate system with a vertical y-axis and origin at the cone's tip, then you want to integrate with respect to y.


This requires you to express the radius of the slab in terms of y. The "thickness" of the slab is dy. The distance that each slab travels to the top also needs to be expressed in terms of y.


To obtain an expression for the radius of the slab, in terms of the slab's height above the origin, consider half a cross-section of the cone (that is, a right triangle). The slab's radius at any height forms a similar right triangle; hence, the slab's radius and the slab's height above the origin are proportional.


In this regard, see whether the diagram below makes sense to you. The radius of the cone is r, and the height of the cone is h.


Also, when a slab is y units above the tip of the cone, how far -- in terms of y -- does it need to travel to reach the top?

slabRadius.JPG


Adapted image; copyright 2012 University of Pennsylvania -- used with assumed permission
 
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mrjust said:
cone with radius [ x= (12-y)/3 ]

integral from 0 to 9 of [ pi( ( 12-y )/3 )^2]62.4*y dy ]
Click to expand...

Ah, I just happened to notice that you responded -- by editing your original post versus adding a new post to the thread. Responding by editing is not a good idea, in general, as the volunteers here rely on the system to notify them of new, unread material. However, when you edit a post, the system does not mark those edits as unread. It's best to show revised work or additional thoughts by adding a new post, so we may see that you've returned.

The radius of the slab is not (12-y)/3. Do you remember how to express a proportion between sides of similar right triangles?

The height that each slab travels to the top of the tank is not y. y is the distance from each slab to the bottom of the tank. Let's think about a specific slab: if this slab is 1 foot from the bottom of the tank, how far does it need to travel to reach the top of the tank?
 
Thanks for your help I do understand that step I just don't know where to go from that. What confusing me is that I don't know how to setup up the integral to obtain the correct answer. To answer your question; It would be 12-y?
 
Thanks for very much for your help. I just solve it.
I changed the variables. 'a' is radius of the slab and 'h' is the height.

Code: 
 Integral from 0 to 9 [ ( h^2/9 )(12-h)62.4*pi ].

I do have a question about the diagram. When I cut out the slab at first I though it was, based on similar triangles, (a/ (12-h) ) = 4/12.
But the correct is (a/h)=4/12. Can you clarify as to why the latter it the correct? I'm having a hard time understading thanks.
 
mrjust said:
To answer your question; It would be 12-y?
Click to expand...

My apologies. I confused myself, by looking at your (12-y)/3 as the radius. Now I recognize your revised integrand as mostly correct because it's about the same as:

integral from y=0 to y=9 [62.4*Pi*(y/3)^2*(y-12) dy]

The only problem is that you were squaring (12-y)/3 instead of squaring y/3.

The integral in blue above gives the result shown in your book, yes?
 
mrjust said:
I do have a question about the diagram. When I cut out the slab at first I though it was, based on similar triangles, (a/ (12-h) ) = 4/12.
But the correct is (a/h)=4/12. Can you clarify as to why the latter it the correct?
Click to expand...

We just cross-posted, heh, heh.

Okay, I'm glad that you solved it.

The reason why the proportion is a/h = 4/12 is that we're comparing corresponding sides of two right triangles.

One triangle has height 12 and base 4. The similar triangle has height y and base a.

We make the comparison using these exact sides.

In English, we say, "4 is to 12 as a is to h".

That statement translates mathematically to: 4/12 = a/h.

Alternatively, we could compare base to base AND height to height: "4 is to a as 12 is to h".

This form gives 4/a = 12/h, and solving this proportion for a yields the same result.

Cheers ~ Mark :cool:
 
mmm4444bot said:
We just cross-posted, heh, heh.

Okay, I'm glad that you solved it.

The reason why the proportion is a/h = 4/12 is that we're comparing corresponding sides of two right triangles.

One triangle has height 12 and base 4. The similar triangle has height y and base a.

We make the comparison using these exact sides.

In English, we say, "4 is to 12 as a is to h".

That statement translates mathematically to: 4/12 = a/h.

Alternatively, we could compare base to base AND height to height: "4 is to a as 12 is to h".

This form gives 4/a = 12/h, and solving this proportion for a yields the same result.

Cheers ~ Mark :cool:
Click to expand...

Lol I took me 6 hours to do this problem. Thanks a lot for your help I really appreciate it Mark.
 
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