Question

Eight sided dice can be described as two square based pyramids stuck together at
the bases. Their faces are equilateral triangles. If the edge of one die is 2mm, find the difference
in materials cost between a solid gold die and a “gold” painted plastic die.
(Gold costs $40.38 per gram. Gold weighs 0.02 grams per cubic mm. Plastic costs $10 per gram.
Plastic weighs .01 grams per cubic mm. “Gold” paint costs $0.01 per square mm.)

Answer 1

It costs $3.63 more for the gold die than the gold plated die.

The volume of each half of the die is given by the formula

V=1/3s³.

This means the total volume is V = 2(1/3s³) = 2(1/3(2³)) = 16/3 = 5.33 mm³.

For the solid gold die, we know that it weighs 0.02g per mm³:

5.33(0.02) = 0.1066g

Gold costs $40.38 per gram:
40.38(0.1066) = $4.30

For the plastic die, plastic weighs 0.01 g per mm³:
0.01(5.33) = 0.0533g

Plastic costs $10 per gram:
0.0533(10) = 0.533 ≈ $0.54.

We also need to find the surface area, as it will be painted.  Each face is an equilateral triangle; the area of a triangle is given by the formula

A = 1/2bh

We need to use the Pythagorean theorem to find the height.  The height will be one leg and 1/2 of the base will be the other; the slanted side will be the hypotenuse:

h² + (1/2(2))² = 2²
h² + 1² = 2²
h² + 1 = 4
h² = 3
h = √3

The area of the triangle would then be A = 1/2bh = 1/2(2)(√3) = √3.  There are 8 such faces on the die, so the total surface area would be 8√3 = 13.86mm².

Paint costs $0.01 per mm²:
13.86(0.01) = 0.1386 ≈ $0.14

Together the cost for the plastic is 0.14+0.53 = $0.67.

The difference between it and the cost for the gold die is 4.30 - 0.67 = $3.63.