consider the following diagram depicting a shoe lace passing through holes. where starting width is 3, and ending width is 1
The width between nodes can either decrease or increase linearly or it can be constant. I'm given the beginning width between the nodes at the top and the ending width at the bottom. the vertical distance (spread) between each pair of nodes is constant. I'm required to determine the entire length of shoe lace.
I know that the change dx from starting width to endwidth is found by the equation
(endWidth - startWidth) / (numPairs - 1);
I tried using the pythagorean method to determine the hypotenuse between the middle pair of nodes and the top pair of nodes then multiplied that by two but that didn't work I assume because that is not a right angled triangle. Also tried using half the width and half the spread to get the hypotenuse of each touching triangle, still didn't work.
Answer
Set up a cartesian coordinate system and explicitly calculate the position of each of the six holes in that coordinate system. Check to make sure the positions you calculate for the six holes match with the vertical distances and the horizontal distances that you know.
Once that is done, you have the $x$ and $y$ coordinates for each hole, and can easily find the distance between any two holes using the Pythagorean theorem. Find the five (not four!) distances and add them up.
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