Question

A triangular pyramid is cut by a plane parallel to its base. The area of the triangle of intersection is only one-third of the base area. If the altitude of the pyramid is 48cm, find the distance of the base of the pyramid from the cutting plane.

Answer 1

i got 20.29cm

Answer 2

what kind of triangle is the base? Does it even matter?

Answer 3

We know that any cut is going to be a similar triangle to the base

Answer 4

The area of the base is: \[
A = kx^2
\]

Answer 5

\[
\frac 12 xy = \frac 13\left(\frac 12 (cy)(cx)\right)
\]

Answer 6

\[
\frac 12 xy = \frac 12 xy \left(\frac{c^2}{3}\right)
\]

Answer 7

\[
\frac{c^2}{3} = 1\implies c= \sqrt{\frac 13}
\]

Answer 8

I think that \(c\) times the pyramid height will be the height from the cross section... so the difference from the base is going to be: \[
h\left(1-\sqrt{\frac 13}\right) = \left(\frac{\sqrt 3-1}{\sqrt 3}\right) 48\text{cm}
\]

Answer 9

http://www.wolframalpha.com/input/?i=%28%28sqrt%283%29-1%29%2F%28sqrt+3%29%2948+cm

Answer 10

\[ \left(\frac{\sqrt 3-1}{\sqrt 3}\right) 48\text{cm}\approx 20.29\text{cm}
\]