A "rectangular" tetrahedron is an irregular tetrahedron with three faces which are right isosceles triangles with legs of length h, and with one face an equilateral triangle with sides √(2)h.

Task

(1) Students are given the value of h, for example, h = 10 cm, and are asked to draw a plan and to construct a rectangular tetrahedron from poster board. Different plans should be considered and discussed.

After they have made their tetrahedrons, they should check that eight of them can be put together to form one regular octahedron.

(2) Next students should compute the volume of the tetrahedron using three different methods.

(a) Using the formula for the volume of a pyramid. When one
of the right triangles is a base, the triangle's area is h^{2}/2, and
the pyramid's height is h. So the volume is 1/3*(area of base)*height,

V
= h^{3}/6

(b) Using an integral. In the same orientation as in (a), let x be the distance along the height from the top toward the base. So the horizontal cross section at level x (from the top) has area,

A(x)
= x^{2}/2

So the integral of A(x) for x changing from 0 to h can be computed on the TI-84 calculator by

fnInt(X^{2}/2,X,0,H)

(c) Using the formula,

V = h*(average(A(x) for 0 ≤ x ≤ h)

It can be approximated by taking different samples of numbers between 0 and h, and

H*mean(H^{2}rand(n)^{2}/2)

is a program that can be used for sample sizes n up to 999.

(You may write it as H*mean((H*rand(N))^{2}/2.)

Remark

H*rand(n) creates a list of length n of numbers x between 0 and H, chosen with a uniform probability distribution.

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