April 19th, 2017, 01:21 PM  #1 
Newbie Joined: Apr 2017 From: Seattle Posts: 14 Thanks: 1  I need help dividing a line into unequal segments!
Here is my problem: Given a line ABCD ABCD.jpg Values I know: AD, AB, CD AB is greater than CD. I need to divide BC into 33 segments. Each segment will decrease in length from AB down to CD. I know it wonâ€™t work if BC is divided up into too many or too few sections. I am confident that given my real world lengths there is enough space to work with. I need to build a formula wherein I can input AD, AB, CD, and the number of divisions for BC. Then have it tell me the resulting lengths of the sections from B to C. I have been using Excel to write out a lot of attempts at linear formulas but typically when all is said and done if I sum the results I get a total length greater than AD. Thank you in advance. 
April 19th, 2017, 02:56 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 19,058 Thanks: 1618 
If the segment lengths, including AB and CD, form an arithmetic progression consisting of n terms (where n > 2), with initial term a = AB and final term l = CD, they total n(a + l)/2, so you need n = 2AD/(AB + CD). In general, n isn't 35, so BC isn't divided into 33 segments. However, your specification is a bit vague and doesn't require such an arithmetic progression. Provided that BC (which is AD  AB  CD) exceeds 33CD, you can do something similar where AB and CD are not included in the arithmetic progression, but the 33 segments that make up BC form an arithmetic progression. Simply choose values for a and l such that AB > a > l > CD and a + l = 2BC/33. 
April 19th, 2017, 05:07 PM  #3 
Newbie Joined: Apr 2017 From: Seattle Posts: 14 Thanks: 1  Thanks and let me elaborate!
I appreciate the reply, but it didn't quite give me an understanding of how to answer the problem. I didn't want to bore you with the actual measurements and application of this math problem but if it helps arrive at the solution easier then so be it! I am making a musical instrument that has 33.3 inches of available length on the soundboard for the strings (represented by AD). There are 36 total strings and therefore 35 spaces between them. I know the first space needs to be 1.18 inches (AB) and the last one needs to be .88 inches (CD). The remaining 33 spaces I need to have go from 1.18 inches down to .88 inches (BC/33) in a nice progression. For all intents I cannot choose new values for a and l (from your explanation) to satisfy the arithmetic. I know it can be frustrating to try and explain these things to people who don't seem to be getting it right away (me ) and I thank you for your patience. I am hoping you will say this is a challenging problem because I have spent 8 hours trying to solve it myself as well as asking the help of no fewer than 3 math teachers. Last edited by skipjack; April 20th, 2017 at 03:31 AM. 
April 20th, 2017, 04:25 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 19,058 Thanks: 1618 
What is your length of BC?

April 20th, 2017, 05:46 AM  #5 
Newbie Joined: Apr 2017 From: Seattle Posts: 14 Thanks: 1 
Ends up being 31.24 inches.

April 20th, 2017, 06:48 AM  #7 
Newbie Joined: Apr 2017 From: Seattle Posts: 14 Thanks: 1 
It's for a harp, not a fretted instrument. Thanks though.

April 20th, 2017, 08:36 AM  #8 
Global Moderator Joined: Dec 2006 Posts: 19,058 Thanks: 1618  Why does CD need to be 0.88 inches? If it were 0.72 inches, it would help, as BC could then be correspondingly longer, with AD remaining unchanged. The difficulty arises because AD/35 = 33.3/35 = 0.95 approximately, but 0.95 is significantly different from the average of 1.18 and 0.88 (whereas it would equal the average of 1.18 and 0.72). 
April 20th, 2017, 09:04 AM  #9 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 12,609 Thanks: 845 
Using simple example: A...3...B..........12..................10........ ......8......C..2..D Givens: AD = 35, AB = 3, CD = 2, BCdivisions = n = 3 Above correct? Question: does "decrease in divisions" need to be a constant, like 2 in above example? 
April 20th, 2017, 09:11 AM  #10 
Newbie Joined: Apr 2017 From: Seattle Posts: 14 Thanks: 1 
skipjack: It needs to be .88 inches because of hardware constraints and playing comfort. Is there just not an equation to figure this out versus hard coding numbers to fit a scheme? Perhaps it is too complicated for a simple formula? Your making me feel better about having spent so much time head scratching! Is there a way to plot this with a curve or some other geometric method of representing it and working an equation back from that? denis: It does not need to be a constant. Just a progression where all the segments go from 1.18 to .88 in 35 steps and all add up to 33.3 inches. Last edited by ronmryd; April 20th, 2017 at 09:28 AM. 

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dividing, line, segments, unequal 
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