Height of IntersectionDate: 01/16/2003 at 21:24:11 From: Rick Subject: Advanced geometry There are two flagpoles with heights of 10 feet and 70 feet respectively that are 100 feet apart on a level surface. If a line ran from the top of each pole to the bottom of the other, what would be the height of intersection? Create a formula using one variable I've concluded through diagrams that the distance apart has nothing to do with the location of the intersection because both the lines are moving proportionately, so the variable must be height. Date: 01/16/2003 at 23:31:59 From: Doctor Peterson Subject: Re: Advanced geometry Hi, Rick. You're right that everything depends not on the actual distance, but simply on proportions. What if we ignore the 100 in the problem, and just use variables for the two horizontal distances and the height of the intersection? (I don't see what the "formula using one variable" might be; but perhaps if we start with three variables, we'll see things simplify.) Here's the setup: + | \ a| \ + | X |b | / h| \ | +-----+---+ x y Here a and b are the two heights, 70 and 10 in the original problem; I like to use variables (in this case, named constants) rather than numbers in problems like this where the specific numbers could easily be replaced with others, because that can give me more insight into the nature of the problem. The height we are looking for is h, and x and y are the horizontal distances. Now I see two pairs of similar triangles. I'll set up two proportions, one relating a to h, and the other relating b to h. That gives two equations, which can then be solved to eliminate x or y. It turns out that both x and y can disappear; if you don't find that happening naturally as you solve the equations, try using each equation separately to get an expression for x/y in terms of a, b, and h, then set those equal. This will give you an equation you can solve for h; in fact, that might be the "formula" you are supposed to find. And it's the fact that only the ratio x/y matters, not the sum x+y, that your insight revealed. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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