Two-base Paraboloid

2 posts / 0 new
Log in or register to post comments
Last post
March 8, 2013 - 12:15pm
#1
Rhozse Manalili's picture
Rhozse Manalili
Offline
Joined: Mar 2 2013 - 2:52pm
Points: 538
Two-base Paraboloid

"The altitude of two-base paraboloid is 8in. and the upper base has an area which is four times the area of the lower base.Find the radii of the solid if the volume is 125pi cube in. "

Vote for this post, click the Like button once
Top
March 8, 2013 - 2:23pm
Romel Verterra's picture
Romel Verterra
Online
Joined: Oct 12 2008 - 3:39pm
Points: 1834
Re: Paraboloid

solid_013-paraboloid-two-base.gif$ A_1 = 4A_2 $

$ \pi R^2 = 4\pi r^2 $

$ R^2 = 4r^2 $

$ R = 2r $
 

$ V = \frac{1}{2}A_1 y_1 - \frac{1}{2}A_2 y_2 $

$ 125\pi = \frac{1}{2}\pi R^2 - \frac{1}{2}\pi r^2 y_2 $

$ 250 = R^2 y_1 - r^2 y_2 $

$ 250 = 4r^2 y_1 - r^2 y_2 $

$ 250 = (4y_1 - y_2)r^2 $
 

By squared property of parabola
$ \dfrac{R^2}{y_1} = \dfrac{r^2}{y_2} $

$ \dfrac{4r^2}{8 + y_2} = \dfrac{r^2}{y_2} $

$ \dfrac{4}{8 + y_2} = \dfrac{1}{y_2} $

$ 4y_2 = 8 + y_2 $

$ 3y_2 = 8 $

$ y_2 = \frac{8}{3} ~ \text{in.} $
 

$ y_1 = 8 + y_2 $

$ y_1 = \frac{32}{3} ~ \text{in.} $
 

Thus,
$ 250 = [ \, 4(\frac{32}{3}) - \frac{8}{3} \, ]r^2 $

$ 250 = 40 r^2 $

$ r = \frac{5}{2} ~ \text{in.} $           answer
 

$ R = 2(\frac{5}{2}) $

$ R = 5 ~ \text{in.} $           answer
 

The volume of paraboloid is given by the formula V = ½πr2h. In the solution, the volume is the difference of paraboloid of radius R and height y1 and paraboloid of radius r and height y2.

up
0 users have voted.
Vote for this answer, click the Thumbs-up icon once
Top
Log in or register to post comments
SPONSORED LINKS
Page copy protected against web site content infringement by Copyscape

Recent comments