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Volume of a Pond

Date: 1 Feb 1995 19:25:49 -0500
From: Richard L. Burnett
Subject: Advice on math problem

I need to have a large pond cleaned at my place of employment.  The pond
is 320' long and is 85' wide.  The pond is 22' deep and the sides
are sloped 2" for every 1' of depth.  The pond is completely filled with
waste material and I would like to know how many cubic yards are in 
the pond to remove.  Thank You for your help!!

Date: 3 Feb 1995 17:38:46 GMT
From: Dr. Math
Subject: Re: Advice on math problem

Dear Richard,

Thanks for writing Dr. Math.  Normally, we take questions only from
students in K-12, but I'll try to give you a few ideas on where to go with
this problem.  

The first thing to do, is to draw a clear picture (this is hard in 3-D
and especially hard on the computer, so I won't even attempt it here!). 
You want to find the volume of the pond.  One way to do this is to
consider the volume of the pond as the difference between the volume of
the pond with no sloped sides (this would simply be 320' x 85' x 22') and
the volume of the parts of the pond that have been filled in.  That is,
find the volume for the space occupied by the "extra" concrete (or sand or
whatever material it is) that makes the pond slope and subtract it from
the total volume of a pond with no sloped sides.  This will get you the
volume you are looking for.

The difficult part is finding the volume for the space occupied by the
extra stuff.  You said the sides are sloped so that for every 1' of depth,
the sides go out by 2".  That means at the bottom of the pond, the sides
have gone out by 44" (44 = 22*2).  So, for instance if we were to look at
a cross section of the pond that is cut perpendicular to one of the sides,
on each side we get a right triangle with legs 22' and 44''.  The volume
along each side is given by the formula:

V  =  bhl/2,   where b is the base and h is the height 
               of the triangle in the cross section, and
               l is the length of the side.

So, the volume along the long side is going to be (44'')(22')(320')/2.  
Note here that some measurements are in inches and others are in feet.  
You'll have to convert so that you are using the same units when you 
actually multiply through to get a numerical answer.  

It might be tempting for us to add up the volumes of the 4 sides and
subtract from the total volume, but there is one more factor to consider. 
Consider a corner of the pond.  If we add up the volumes along each side,
part of the volume of the filled in part of the pond will be counted
twice.  (This will become clear from a picture).  So, we need to figure
out how to account for this volume.  I think this is a very tricky
problem, indeed, and I will tell you my first instinct, though it may be
wrong.  You tell me if you think it makes sense!  Let's consider the
volume of the filled in portion of the pond just at the corners, more
precisely, the volume of portion that has a square base of 44''.  It is in
this area that the duplications are made.  We could try to figure out how
much of a duplication is made, or we can try to figure out the volume of
this area directly.  It seems to me, that perhaps we might want to use a
triple integral here to find the volume.  How familiar are you with
multi-variable calculus?  Since I'm not sure how much calculus you have
had, why don't you think about this problem for a bit, see if you come up
with any good methods for finding the volume of this portion of the pond,
and write back if you have any more problems.  If you tell me your
familiarity with calculus, I'd be happy to tell you about how you might
use it to find the volume of this trouble spot.  There may be an easier
way to find this volume that I'm overlooking...

   Hope this helps!

--Sydney, "dr. math"

Date: 3 Feb 1995 18:25:13 GMT
From: Dr. Math
Subject: Re: Advice on math problem

Dear Richard,

There is also another way to do your problem that will make things much,
MUCH easier.  Do you know how to find the volume of a prism?  You could
consider your pond to be a prism that has had the top cut off.  I don't
have the formula for the prism right here, but perhaps you do.  If you
have any questions about this method or the other method I wrote about
earlier, please feel free to write back.

--Sydney, "dr. math"  
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry
High School Practical Geometry

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