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### Making a Pool Tarp

```
Date: 9/20/95 at 14:26:7
From: Anonymous
Subject: geometric slope

I am trying to construct a pool frame out of PVC that will be
placed over a pool.  We already have the pool tarp, but we need
to build something that will shed water so it has to have a
slope.  PVC only comes in 90 degree and 45 degree angles and
straight sections.  How do I figure out the maximum height the
frame can stick straight up in the air to utilize the tarp.
The pool tarp is:

272.5 inches width
484 inches length

The pool frame (that needs to be constructed out of pvc can
not be smaller than):

226 inches
412 inches

How do figure out the maximum height based on the pool cover
dimensions so that the cover will fit on the frame width wide.
I can make up for the length but the width is crucial.... (this
is difficult to explain a drawing would be better).

How do I figure out if this will even work?  HELP!!!
```

```
Date: 9/22/95 at 16:22:5
From: Doctor Andrew
Subject: Re: geometric slope

If I understand your problem correctly, you want to build
something that looks like this (not as steep though) to cover

/|\      -
/ | \     |
z  |  z    h
/   | A \   |
/    |    \  |
|~~~~~~~~~|  -
|  pool   |
\________/
|-x-|

You want to have the biggest height h possible in order to
have a steepest slope possible on your pool frame.  Let x be
half the width of the base of the frame.  Let z be half the
width of the tarp. It turns out, as might be intuitive, that
constructing the frame symmetrically across the pool yields the
largest h.  (You can check this with a piece of paper or
string.)

The Pythagorean Theorem states that for a right triangle (one
with two perpendicular sides), like the triangle marked "A" in
the diagram, the sum of the squares of the two smaller sides
equals the square of the longer side.  So,

z^2 = x^2 + h^2

Now, we need to solve for h in terms of x and z.  Well,
subtracting x^2 from both sides of the equation we get

h^2 = z^2 - x^2.

Taking the square root of each side, we get

h = square root(z^2 - x^2).

In order to make h as big as possible we want the largest z and
smallest x we can use.  The largest z is half the width of the
tarp.  The smallest x is half the width of the smallest base of
the frame we can use.

So, in your case, h = square root ((272.5 / 2)^2 - (226 /
2)^2) = 76.1253 in.

You can also check to make sure you have enough length using
the same technique.  If width is ignored, the optimal h for the
length of the tarp is

h = square root ((484 / 2)^2 - (412 / 2)^2) = 126.996 in.

So, it looks like width is, as you already said, the limiting
factor.

Before you go cutting any PVC you could cut yourself a 272.5
inch and a 226 inch piece of string.  If you pin the shorter
one at each end, the string should be able to fit a 76.1253
inch stick standing upright between the ground and the longer
string.

string experiment in case I've made a mistake. Good luck!

-Doctor Andrew, The Geometry Forum
```

```
Date: 9/27/95 at 13:28:25
Date: Wed, 27 Sep 1995 13:28:54 -0500
From: ELIZABETH A REYNOLDS

I appreciate your math!  I will do what you say and try it
with string first.... but it does seem to me that 76 inches
is a HUGE amount of height considering that the overlap
between the pool tarp width and the pool frame is 46.5
inches.  I don't see how you could have a height of apx 8
ft.... something isn't right with the equation.... do you
have any ideas???

P.S. Your tarp drawing is correct.  I just don't think that
eight feet high would be the correct answer if the overlap is
only 46.5 inches in the width.
```

```
Date: 9/27/95 at 14:58:38
From: Doctor Andrew

It sure seems huge, but I've checked it.  It certainly isn't
intuitive.  There is another problem where you have a steel
belt around the equator of a earth and you add a foot to it.
Suddenly it is off the face of the earth by miles if I
remember correctly.  Anyway, with regard to this problem, you

If 113 if half the frame width and 76 is the maximum height:

sqrt(113^2 + 76^2) = 136

136 is half the tarp, right?

So it seems to work.  When I worked on your problem I didn't
really stop to think about the physical implications of the
result, but it's pretty neat.

I have a feeling you're not going to be building a 6 ft pool
frame, but it sure looks like you have enough tarp.  Still,
I'd try it with the string. :^)

- Doctor Andrew, The Geometry Forum

```
Associated Topics:
High School Geometry
High School Practical Geometry
High School Triangles and Other Polygons

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