Mock AIME 1 Pre 2005 Problems/Problem 8
Problem
, a rectangle with
and
, is the base of pyramid
, which has a height of
. A plane parallel to
is passed through
, dividing
into a frustum
and a smaller pyramid
. Let
denote the center of the circumsphere of
, and let
denote the apex of
. If the volume of
is eight times that of
, then the value of
can be expressed as
, where
and
are relatively prime positive integers. Compute the value of
.
Solution
As we are dealing with volumes, the ratio of the volume of to
is the cube of the ratio of the height of
to
.
Thus, the height of is
times the height of
, and thus the height of each is
.
Thus, the top of the frustum is a rectangle with
and
.
Now, consider the plane that contains diagonal as well as the altitude of
. Taking the cross section of the frustum along this plane gives the trapezoid
, inscribed in an equatorial circular section of the sphere. It suffices to consider this circle.
First, we want the length of . This is given by the Pythagorean Theorem over triangle
to be
. Thus,
. Since the height of this trapezoid is
, and
extends a distance of
on either direction of
, we can use a 5-12-13 triangle to determine that
.
Now, we wish to find a point equidistant from ,
, and
. By symmetry, this point, namely
, must lie on the perpendicular bisector of
. Let
be
units from
in
. By the Pythagorean Theorem twice,
Subtracting gives
. Thus
and
.
See also
Mock AIME 1 Pre 2005 (Problems, Source) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |