My Math Forum  

Go Back   My Math Forum > College Math Forum > Calculus

Calculus Calculus Math Forum


Reply
 
LinkBack Thread Tools Display Modes
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.
angofango87 is offline  
 
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^3-50.84z^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.
shai is offline  
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?
angofango87 is offline  
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.
shai is offline  
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..
Attached Images
File Type: jpg cone1.jpg (8.2 KB, 5 views)

Last edited by skipjack; February 18th, 2016 at 05:20 AM.
shai is offline  
Reply

  My Math Forum > College Math Forum > Calculus

Tags
changing, cone, height, radii



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Geometry - Reducing Cone Volume by Height. Oshikorosu Geometry 9 March 31st, 2013 08:08 PM
Radii of Convergence veronicak5678 Complex Analysis 8 March 14th, 2012 05:32 PM
Define the relationship of radii DocBane Algebra 1 March 16th, 2010 03:33 AM
Why is the Groebner cone a cone? nightingale Abstract Algebra 0 February 19th, 2010 04:24 AM
Cone + Cone Frustum Formula Silvester Algebra 0 February 17th, 2009 09:29 AM





Copyright © 2019 My Math Forum. All rights reserved.