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October 23rd, 2019, 09:41 AM  #1 
Math Team Joined: Jul 2013 From: काठमाडौं, नेपाल Posts: 911 Thanks: 64 Math Focus: सामान्य गणित  General expression for the total length
i want to derive an expression to calculate the total length of the green lines in the following figure. 1. Firstly we have a Lshape frame of green lines. 2. Horizontal green lines are places inside the frame starting from the bottom, each "D" distance apart. 3. Vertical green lines are placed inside the frame starting from the left, each "D" distance apart. 4. Arrow shows the direction how the lines are placed. 5. The last vertical/horizontal green line is "D or less" distance from the green line that forms the Lshape frame. 
October 23rd, 2019, 11:06 AM  #2 
Senior Member Joined: Jun 2019 From: USA Posts: 386 Thanks: 211 
N1 = number of lines of length L1 = $\displaystyle \left \lfloor \frac{W2W1}{D}+1 \right \rfloor$ N2 = number of lines of length (L1L2) = $\displaystyle \left \lfloor \frac{W2}{D}  N1 + 2 \right \rfloor$ N3 = number of lines of length W1 = $\displaystyle \left \lfloor \frac{L2L1}{D}+1 \right \rfloor$ N4 = number of lines of length (W1W2) = $\displaystyle \left \lfloor \frac{L2}{D}  N3 + 2 \right \rfloor$ Plus one wire of length L2 and one wire of length W2. Enough information for you? 
October 23rd, 2019, 12:12 PM  #3 
Math Team Joined: Jul 2013 From: काठमाडौं, नेपाल Posts: 911 Thanks: 64 Math Focus: सामान्य गणित 
I also agree with N1 and N3 (beside you have interchanged L1&L2 and W1&W2) N2 = ceil(W1/D +1  N1)? N4 = ceil(L1/D + 1  N3)? 
October 23rd, 2019, 01:05 PM  #4 
Senior Member Joined: Jun 2019 From: USA Posts: 386 Thanks: 211 
I think that works out to be the same thing, doesn't it? Check against your sample picture.

October 23rd, 2019, 06:57 PM  #5 
Math Team Joined: Jul 2013 From: काठमाडौं, नेपाल Posts: 911 Thanks: 64 Math Focus: सामान्य गणित 
The sample picture is just a representation. The last spacing can be "D" or "less than D". That way floor function will change the picture. Please read the description.

October 23rd, 2019, 08:53 PM  #6 
Senior Member Joined: Jun 2019 From: USA Posts: 386 Thanks: 211 
Ah, you're right. (To be fair, I understood the setup, I just... can't count, apparently.)

October 23rd, 2019, 09:03 PM  #7 
Math Team Joined: Jul 2013 From: काठमाडौं, नेपाल Posts: 911 Thanks: 64 Math Focus: सामान्य गणित 
I also discovered something. I think we cannot add those L2 and W2 in cases where, the last distance between green line and the frame is exactly "D" What's your opinion on this ? 

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expression, general, length, total 
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