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.)
find the volume of 2 pyramids. 2* 1/3 Bh
multiply by gold cos/mm^3
find area of 8 triangles. 8* 1/2 Bh
multiply by paint cost/mm^2
54.96
To find the difference in material cost between a solid gold die and a "gold" painted plastic die, we need to determine the volume of each die and then calculate the cost based on the given prices.
Let's start with the solid gold die:
1. Determine the volume of one solid gold die:
Since each face of the die is an equilateral triangle, and the edge of the die is given as 2mm, we can use the formula for the volume of a square-based pyramid.
The volume of one pyramid is given by: V = (1/3) * A * h, where A is the area of the base and h is the height.
The area of an equilateral triangle is given by: A = (√3 / 4) * a^2, where a is the length of the side.
The height of the pyramid is equal to the edge length, which is 2mm.
Substituting the values:
A = (√3 / 4) * (2mm)^2 = (√3 / 4) * 4 mm^2 = (√3) mm^2
V = (1/3) * (√3) mm^2 * 2mm = (2/3) (√3) mm^3
2. Convert the volume to grams:
Given that gold weighs 0.02 grams per cubic mm:
Mass of gold = Volume * Density = (2/3) (√3) mm^3 * 0.02 g/mm^3
3. Calculate the cost of the solid gold die:
Given that gold costs $40.38 per gram:
Cost of solid gold die = Mass of gold * Price per gram = (2/3) (√3) mm^3 * 0.02 g/mm^3 * $40.38/g
Now, let's move on to the "gold" painted plastic die:
1. Determine the volume of one "gold" painted plastic die:
We have two square-based pyramids stuck together at the bases, so the total volume will be twice the volume of one pyramid (as calculated earlier).
Volume of "gold" painted plastic die = 2 * V = 2 * (2/3) (√3) mm^3
2. Convert the volume to grams:
Given that plastic weighs 0.01 grams per cubic mm:
Mass of plastic = Volume * Density = 2 * (2/3) (√3) mm^3 * 0.01 g/mm^3
3. Calculate the cost of the "gold" painted plastic die:
Given that plastic costs $10 per gram:
Cost of "gold" painted plastic die = Mass of plastic * Price per gram = 2 * (2/3) (√3) mm^3 * 0.01 g/mm^3 * $10/g
Now, we can find the difference in material cost by subtracting the cost of the "gold" painted plastic die from the cost of the solid gold die:
Difference in material cost = Cost of solid gold die - Cost of "gold" painted plastic die
Please input the values for √3, a, and the given prices to calculate the final difference in material cost.