Math Is Fun Forum

MoyMarian

Member

Registered: 2009-01-10

Posts: 29

solid mensuration III

a pyramid of altitude a and a square base of length of the side b is divided into three parts by two planes passed parallel to the base. If these planes are distant 1/3a and 2/3a, respectively, from the vertex, find the ratio of the volume of the middle part of the volume of the largest part.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: solid mensuration III

The base has area b², so the whole pyramid has a volume of ab²/3.
The shape above the lower plane is a smaller version is this pyramid, scaled by 2/3.
The volume of this "medium-sized" pyramid is (2/3)³ x ab²/3.

There's also a small pyramid above the upper plane, whose volume is (1/3)³ x ab²/3.

Now, the region below the lower plane is the large pyramid minus the medium pyramid.
Similarly, the region between the two planes is the medium pyramid minus the small one.

You can work out their volumes that way, and from them you can get the ratio.

---

Why did the vector cross the road?
It wanted to be normal.