Question

a toy manufacturer has designed a new piece for use in building models. It is a cube with side lengths 7 mm and has a 3 mm in diameter circular hole cut through the middle. The manufacturer wants 1, 000, 000 prototypes . It's a plastic used to create the piece costs $207 per cubic meter, how much will the prototypes cost?

Answer 1

The cost of the prototypes is $ 60.76

How to find the cost of the prototypes

To calculate the cost of producing 1,000,000 prototypes of the cube with a hole, we need to find the volume of one prototype and then multiply it by the number of prototypes and the cost per cubic meter.

The cube's volume is given by the formula

= s³

= 7³

= 343 cubic mm

Volume of the cylindrical hole

= π * (d/2)² * h

= π * (3/2)² * 7

= 49.48 cubic mm

The total volume of one prototype

= 343 - 49.48

= 293.52 cubic mm

convert to cubic meter

= 293.52 * 10⁻⁹

The total cost:

= 293.52 * 10⁻⁹ * 1,000,000 * $ 207

= $ 60.76

Answer 2

Given that the cubical piece has side (s) 7 mm, and a cylindrical hole of diameter (d) as 3 mm.

The volume of the piece can be calculated as,

`\begin{gathered} \text{Volume of piece}=\text{ Volume of cube}-\text{ Volume of cylinder} \\ V=s^3-(\pi)/(4)d^2s \\ V=7^3-(\pi)/(4)(3)^2(7) \\ V=343-(63\pi)/(4) \\ V\approx293.52mm^3 \\ V\approx293.52*10^(-9)\text{ }m^3 \end{gathered}`

It is mentioned that the material costs $207 per cubic meter, so the cost (c) of 1 piece is calculated as,

`\begin{gathered} \text{Cost of 1 piece}=\text{ Cost per unit volume}*\text{ Volume of 1 piece} \\ c=207*293.52*10^(-9) \\ c\approx60758.64*10^(-9) \end{gathered}`

This is the cost (in dollars) for 1 piece.

Given that the manufacturer wants to produce 1,000,000 such pieces, so the total cost (TC) is calculated as,

`\begin{gathered} \text{Total Cost}=\text{ Cost per piece}*\text{ No. of pieces} \\ TC=60758.64*10^(-9)*1,000,000 \\ TC=60758.64*10^(-9)*10^6 \\ TC=60758.64*10^(-3) \\ TC=60.75864 \\ TC\approx60.75 \end{gathered}`

Thus, the total prototypes will cost around $60.75