A 'Pyramiddle' Tent Problem
Date: 07/12/99 at 17:13:46 From: Hank Hufnagel Subject: Algebra Hi, I am a scoutmaster, and, in the course of designing an old-fashioned kind of tent, I have bumped into an algebra problem that is beyond me. You can find details at: Hank's Tent Problem http://www.hufsoft.com/bsa51/page7.html "...Making liberal use of the Pythagorean Theorem, the four right triangles give the equations: x^2 = d^2 + h^2 y^2 = d^2 + 4^2 z^2 = (r-d)^2 + (h-d)^2 y^2 = x^2 + z^2 "The idea was then to solve for d in terms of h and r. This would let me design a pyramiddle tent from nearly any rectangular tarp... "Can you solve the problem? Just figure out an equation that yields d when values for h and r are inserted." It would be great if you or someone you know could crack this for me. Thanks, Hank Hufnagel
Date: 07/13/99 at 15:31:38 From: Doctor Rick Subject: Re: Algebra Hi, Hank. That's a nice problem! A practical real-life problem with some interesting geometry and plenty of Pythagorean Theorem exercise, plus solution of multiple simultaneous equations and solution of a quadratic. I hope math teachers will get to use it! Here is how to solve the problem. Your 4 equations are: (1) x^2 = d^2 + h^2 (2) y^2 = d^2 + r^2 (3) z^2 = (r-d)^2 + (h-d)^2 (4) y^2 = x^2 + z^2 With r and h given, we have 4 unknowns: d, x, y, and z. First eliminate y by equating (2) and (4): (5) d^2 + r^2 = x^2 + z^2 Next, use (1) to substitute for x^2 in (5): d^2 + r^2 = d^2 + h^2 + z^2 Subtract d^2 from both sides: r^2 = h^2 + z^2 Now we can solve for z (which is an important quantity, the depth of the tent) in terms of the given r and h: (6) z^2 = r^2 - h^2 Use (6) to substitute for z^2 in (3): r^2 - h^2 = (r-d)^2 + (h-d)^2 r^2 - h^2 = r^2 - 2rd + d^2 + h^2 - 2hd + d^2 d^2 - 2d(r+h) + h^2 = 0 Now we have a quadratic equation. We can use the quadratic formula to solve it: d = (r + h +- sqrt((r+h)^2 - 4h^2))/2 Since d can't be greater than r or h, only the negative sign makes sense. We can rewrite the square root a little, and we get: d = (r+h - sqrt((r+3h)(r-h)))/2 There is your formula; it gives the results you have in your table. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/
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