A toy manufacturer has designed a new piece for use in building models . It is a cube with side length 7 mm and it has a 3 mm diameter circular hole cut through the middle. The manufacture wants 1,000,000 prototypes. If the plastic used to create the piece costs $270 per cubic meter, how much will the prototypes cost? Be exact.
To find the volume of the cube, we need to raise the side length to the power of 3: 7 mm * 7 mm * 7 mm = <<7*7*7=343>>343 mm^3
The hole in the middle doesn't change the volume of the cube, so we can ignore it.
To convert from mm^3 to cubic meters, we need to multiply by the conversion factor 1 meter / 1,000,000 mm cubed: 343 mm^3 * 1 m / 1,000,000 mm^3 = 0.000343 m^3
To find the cost of one prototype, we need to multiply the volume of the cube by the cost per cubic meter: 0.000343 m^3 * $270 / m^3 = $92.61.
To find the total cost of the prototypes, we multiply the cost per prototype by the number of prototypes: $92.61 per prototype * 1,000,000 prototypes = $<<92610000=92610000>>92,610,000. Answer: \boxed{92,610,000}.
To calculate the cost of the prototypes, we need to determine the volume of each cube and then multiply it by the number of prototypes.
1. Calculate the volume of the cube:
The volume of a cube is given by the formula V = side length^3. In this case, the side length is 7 mm.
V = 7 mm * 7 mm * 7 mm
V = 343 mm^3
2. Convert the volume to cubic meters:
Since the cost is given in terms of cubic meters, we need to convert the volume to cubic meters.
1 m = 1000 mm (since 1 meter is equal to 1000 millimeters)
V = 343 mm^3 * (1/1000 m/mm)^3
V = 343 mm^3 * (1/1000)^3 m^3/mm^3
V = 343 mm^3 * (1/1000)^3 m^3/mm^3
V = 0.000000343 m^3
3. Calculate the total volume of all prototypes:
Since we need 1,000,000 prototypes, we need to multiply the volume of each cube by the number of prototypes.
Total Volume = 0.000000343 m^3 * 1,000,000
Total Volume = 343 m^3
4. Calculate the cost of the prototypes:
The cost is given as $270 per cubic meter, so we need to multiply the total volume by the cost per cubic meter.
Cost = 343 m^3 * $270/m^3
Cost = $92,610
Therefore, the prototypes will cost $92,610.