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February 17th, 2016, 02:48 PM  #1 
Newbie Joined: Feb 2016 From: Australia Posts: 2 Thanks: 0  Changing radii and height in a cone
Hi, We have to design a depth gauge for a container but they come in different shapes and sizes. Generally they have a cylindrical shape with the diameters of the top and bottom being different. Short of filling each container with known volumes of water and measuring the height, how could I derive equations then plug them into a spreadsheet so if I know the two diameters and the full volume, I can determine the height change in order to mark the depth gauge. I would like to mark the gauge with 2.5 litre increments Top radius (r1) = 301 mm Bottom radius (r2) = 225 mm Height when full (h) = 450 mm I think the variables I need to determine are dr/dV and dh/dV. Thoughts? Last edited by angofango87; February 17th, 2016 at 02:56 PM. 
February 18th, 2016, 01:05 AM  #2 
Newbie Joined: Feb 2016 From: israel Posts: 17 Thanks: 3 
r(z)=0.168z+301 [mm], z is the current height of the water inside V=pi*int[r(z)^2]dz If I didn't make a mistake, you should get V= pi(0.0095z^350.84z^2+90601z) [mm^3]. You can make an Excel chart of V=2.5, 5, 7.5.... and you will have the matching height for each volume increment. Divide by 1000 to work with cm^3. Last edited by skipjack; February 18th, 2016 at 05:15 AM. 
February 18th, 2016, 01:22 AM  #3 
Newbie Joined: Feb 2016 From: Australia Posts: 2 Thanks: 0 
Thanks Shai, I am having some trouble solving for z when V is known. This is what I need for the spreadsheet. Maybe if we did it in meters that might also help when converting. Could you please explain how you got the constants in this case? 
February 18th, 2016, 03:50 AM  #4 
Newbie Joined: Feb 2016 From: israel Posts: 17 Thanks: 3 
Since the container has a truncated cone shape, its radius changes linearly. r(z)= A*z+B The "boundary conditions" are r(0)=301, r(450)=225. Substitute and find A,B. A=pi*r(z)^2 and V is the integral of A(z)dz. You can try and plot this equation (MAPLE or MATLAB?). Last edited by skipjack; February 18th, 2016 at 05:20 AM. 
February 18th, 2016, 04:52 AM  #5 
Newbie Joined: Feb 2016 From: israel Posts: 17 Thanks: 3  see attached
I've attached a Maple plot..
Last edited by skipjack; February 18th, 2016 at 05:20 AM. 

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changing, cone, height, radii 
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