Making a Pool TarpDate: 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 your pool: /|\ - / | \ | 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. I sure hope I've answered the right question. Please try the 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 Subject: Re: geometric slope -Reply 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 Subject: Re: geometric slope -Reply 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 can just use the Pythagorean Theorem to check your answer: 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 |
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