Tuesday 31 October 2017

geometry - I have X feet of rope. What should maximum size of triangle be?

Since we're approaching the Christmas season, I'm calculating how many feet of lights I need for a few decorations.



Let's say I have X feet of lights, is it possible to calculate the height/width of an isosceles triangle so that it's outlined completely in lights? I can always divide the isosceles triangle in two right triangles, and then cover each one separately.



So in terms of angles, the outer rectangle should either be 70, 90, & 20 degrees (if divided into two rectangles) or 70, 90, and 40.



Three questions:

If I have X feet of string, how high & wide should the isosceles triangle be so that it's covered completely by the string?



If I have X feet of string, how high & wide should the right triangle be so that it's covered completely by the string?



Instead of using one piece of string for the isosceles triangle (which has 6 sides total), would I use less string if I divide it in two right triangles? I would need 2 strings, but maybe the sum of these two strings is less than the one string I would use if I covered the isosceles triangle.



Anyways I can buy the string in 30 or 60 feet. And if I buy the lights, I don't want to be short or over-buy. And since they're a specific length, I want the triangle to use the as much of the lights (ie. not buy a 60' string so that the whole triangle only uses 50').



Thanks.
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