My Math Forum  

Go Back   My Math Forum > College Math Forum > Calculus

Calculus Calculus Math Forum


Reply
 
LinkBack Thread Tools Display Modes
October 22nd, 2014, 01:06 PM   #1
Member
 
Joined: Oct 2013

Posts: 38
Thanks: 2

Pulling up water from an Inverted Cone - a few points of confusion

Hi all,

Given an inverted cone tank of water (full), the radius is 1.5m at the top, and the length, from top to bottom is 12m - We need to measure the work from pulling up all the water. Density and gravity are just standard here (1000, 9. eight [posted a smiley face when I put "eight"]).

I have two points of confusion here:

1) When finding out the radius, I set up similar triangles. 1.5/12 = r/12-y. This is incorrect, because you get a y value, and a constant. The exercise uses just plain old point-slope, with the bottom of this tank at the origin. I see what the correct method is, but I don't understand why my use of sim. triangles here is incorrect.

*The correct answer for the radius is y/8.

2) Once I have the area for the circle (pi y^2 / 64), my instinct is to sum up the areas of all the circles, when in fact we have to multiply that by 12-y. I am confused as to why we can't sum up the circles with pi y^2 / 64, as well as, obviously, the constants.

Thanks for looking.

Last edited by leo255; October 22nd, 2014 at 01:09 PM.
leo255 is offline  
 
October 22nd, 2014, 01:40 PM   #2
Math Team
 
skeeter's Avatar
 
Joined: Jul 2011
From: Texas

Posts: 3,002
Thanks: 1588

work =$\displaystyle \int$ WALT

W = weight density
A = cross-sectional area of a horizontal slice of liquid w/r to y
L = lift distance of a slice in terms of y
T - slice thickness (dy)

using a set of coordinate axes, sketch a line from the origin (bottom of cone) to the point (1.5,12) ... equation of this line is

$\displaystyle y = 8x \implies x = \frac{y}{8}$

this line forms the cone when rotated about the y-axis.

$\displaystyle r = x = \frac{y}{8}$

$\displaystyle A = \pi x^2 = \pi \left(\frac{y}{8}\right)^2$

$\displaystyle L = 12-y$

work = $\displaystyle W\pi \int_0^{12} \left(\frac{y}{8}\right)^2(12-y) \, dy$

Last edited by skeeter; October 22nd, 2014 at 02:15 PM.
skeeter is online now  
Reply

  My Math Forum > College Math Forum > Calculus

Tags
cone, confusion, inverted, points, pulling, water



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
5+16/3x=5/9 Pulling my hair out over this. Help, please. Mmobley15 Algebra 1 February 23rd, 2014 09:19 PM
1 gram of water 10C + 1 calorie of heat = 1 gram of water @. r-soy Physics 1 December 15th, 2013 02:54 PM
A water tank has shape of an inverted circular cone of altit urduworld Calculus 2 November 1st, 2009 03:06 AM
water is poured into an inverted circular cone of base radiu urduworld Calculus 2 October 31st, 2009 05:31 AM
Cone + Cone Frustum Formula Silvester Algebra 0 February 17th, 2009 09:29 AM





Copyright © 2019 My Math Forum. All rights reserved.