Expert: Scotto - 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

lets say dimensions of the pyramid is 1m in height and 1m by 1m on the base

Answer
The volume of a pyramid is Ah/3 where A is the area of the base and h is the height.  The area of the base is BC where B is the length of one side and C is the length of the other.

Since we are given A, B, and C are all 1, the volume if 1/3.

In order for the area of the base to be made to be 1/2 of the original area of the base, the sides must be 1/√2 as long as they were.  Doing this would make the area of the base into (1√2)(1/√2).
Multiplying this out gives 1/2, which is half of the original.

The ralation between the height and the length of each side of the pyramid is linear.  This means the height will go to 1/√2 as well.

The volume is then changed by putting in 1/2 as the area of the base and 1/√2 as the new height into the volume formula of a pyramid.

The volume formula was given at the start as V = Ah/3.

It is good to remember that the new area A was made to be 1/2 and
the new height is 1/√2.