Question

A toy manufacture has designed a new part for use in building models. The part is a cube with side length 14 mm and it has a 12 mm diameter circular hole cut through the middle. The manufacture wants 9,000 prototypes. If the plastic used to create the part costs $0.07 per cubic millimeter, how much will the plastic for the prototypes cost?

Answer 1

Explanation:

The first step is to find the volume of the cube with the circular hole in the middle. The volume of the cube can be calculated as:

V_cube = (side length)^3 = 14^3 = 2744 mm^3

The hole is a cylinder with radius 6 mm and height 14 mm (the same as the side length of the cube). The volume of the cylinder can be calculated as:

V_cylinder = π(radius)^2(height) = π(6^2)(14) ≈ 1,657 mm^3

Therefore, the total volume of plastic used in each prototype is:

V_total = V_cube - V_cylinder ≈ 1,087 mm^3

To find the total volume of plastic needed for 9,000 prototypes, we can multiply the volume per prototype by the number of prototypes:

V_total_9000 = 9,000 * V_total ≈ 9,783,000 mm^3

Finally, we can calculate the cost of the plastic by multiplying the total volume by the cost per cubic millimeter:

Cost = V_total_9000 * $0.07/mm^3 ≈ $685,810

Therefore, the plastic for the prototypes will cost approximately $685,810.