Expert: Paul Klarreich - 8/30/2010

Question
Hi there were studying volumes of pyramids and while practice solving this one i couldnt get. Any help would be awesome thanks, the problem goes like this

what fraction of the volume of a pyramid must be cut off by a plane parallel to the base if the pyramid thus form has a lateral area equal to one-half of the lateral area of the original pyramid?
answer in the book says .35355 just as a guide

Answer
Questioner:   Francis
Category:  Calculus
Private:  no
 
Subject:  pyramids
Question:  Hi there were studying volumes of pyramids and while practice solving this one i couldnt get. Any help would be awesome thanks, the problem goes like this what fraction of the volume of a pyramid must be cut off by a plane parallel to the base if the pyramid thus form has a lateral area equal to one-half of the lateral area of the original pyramid? answer in the book says .35355 just as a guide

Assume:

1. the pyramid has four sides.
2. the base is a square of side = s.
3. the height  = h.

Then the lateral area is four times each triangle.

Each triangle = 1/2 sh

LA = 4 * that = 2 sh


The volume = 1/3 base * height.

The base is the square s^2.

V = 1/3 s^2 h

...................
Now we cut it at a point that is some fraction, x, of the whole height.

Then the smaller pyramid has base with side  xs, and
    the smaller pyramid has height  xh, and

Its new lateral area is  2 (xs)(xh) = 2x^2 sh.

If this is one-half the original LA, then

2x^2 sh = 1/2 (2 sh)

2x^2 sh =  sh

2x^2  =  1

x^2 = 1/2
x = 1/ sqrt(2)

Now the volume of the smaller pyramid will be:

v = 1/3 base * height. = 1/3(xs)^2 xh

v =  1/3 x^3 s^2h

Now the fraction of the original volume:

v     1/3 x^3 s^2h
--- = -----------------
V       1/3 s^2h

= x^3 = 1/(sqrt(2))^3

= 1/sqrt(8) = (you guessed it) 0.35355339059327376220042218105242