An artisan has 63 kg of metal with a density of [tex]$7000 \, \text{kg/m}^3$[/tex]. He intends to use it to make a rectangular prism with external dimensions of 12 cm by 15 cm. Calculate the length.

Sure! Let's break down how to find the height of the rectangular piece of metal.

1. Understanding the Problem:
- We have a piece of metal that weighs 63 kg.
- The density of the metal is given as 7000 kg/m³.
- The rectangular piece of metal has external dimensions of 12 cm by 15 cm.
- We need to figure out the height of this rectangular piece.

2. Concepts to Use:
- Density is defined as mass divided by volume. We can rearrange this formula to calculate the volume of the metal: Volume = Mass / Density.
- The volume of a rectangular prism (like our metal piece) can also be computed by multiplying its length, width, and height: Volume = Length × Width × Height.

3. Calculation Steps:

- Step 1: Find the Volume of the Metal
- Use the formula for volume: Volume = Mass / Density
- Given: Mass = 63 kg, Density = 7000 kg/m³
- Volume = 63 kg / 7000 kg/m³ = 0.009 m³

- Step 2: Convert Length and Width from Centimeters to Meters
- Length = 12 cm = 12 / 100 meters = 0.12 meters
- Width = 15 cm = 15 / 100 meters = 0.15 meters

- Step 3: Calculate the Height of the Rectangular Piece
- Using the formula for the volume of a rectangular prism: Volume = Length × Width × Height
- Rearrange to find Height: Height = Volume / (Length × Width)
- Plug the known values: Height = 0.009 m³ / (0.12 m × 0.15 m) = 0.5 meters

- Step 4: Convert Height from Meters to Centimeters
- Height = 0.5 meters = 0.5 × 100 centimeters = 50 centimeters

So, the height of the rectangular metal piece is 50 centimeters.