Question
a cube of steel has a mass of 1,000 kilograms. What are the dimensions of the cube if the density of steel is 8,000 kg/m^3
Answers
GPT-4o mini
To find the dimensions of a cube of steel given its mass and density, we can use the relationship between mass, volume, and density. The formula is:
\[
\text{Density} = \frac{\text{Mass}}{\text{Volume}}
\]
Rearranging this formula to find the volume gives us:
\[
\text{Volume} = \frac{\text{Mass}}{\text{Density}}
\]
Given:
- Mass = 1,000 kg
- Density = 8,000 kg/m³
We can now calculate the volume:
\[
\text{Volume} = \frac{1,000 \, \text{kg}}{8,000 \, \text{kg/m}^3} = 0.125 \, \text{m}^3
\]
Since it is a cube, the volume can also be expressed in terms of the length of a side \(s\):
\[
\text{Volume} = s^3
\]
Equating the two expressions for volume gives us:
\[
s^3 = 0.125 \, \text{m}^3
\]
Now, we can solve for \(s\):
\[
s = \sqrt[3]{0.125} = 0.5 \, \text{m}
\]
So, the dimensions of the cube are:
\[
\text{Length} = \text{Width} = \text{Height} = 0.5 \, \text{m}
\]
Thus, each side of the cube measures \(0.5 \, \text{m}\).
\[
\text{Density} = \frac{\text{Mass}}{\text{Volume}}
\]
Rearranging this formula to find the volume gives us:
\[
\text{Volume} = \frac{\text{Mass}}{\text{Density}}
\]
Given:
- Mass = 1,000 kg
- Density = 8,000 kg/m³
We can now calculate the volume:
\[
\text{Volume} = \frac{1,000 \, \text{kg}}{8,000 \, \text{kg/m}^3} = 0.125 \, \text{m}^3
\]
Since it is a cube, the volume can also be expressed in terms of the length of a side \(s\):
\[
\text{Volume} = s^3
\]
Equating the two expressions for volume gives us:
\[
s^3 = 0.125 \, \text{m}^3
\]
Now, we can solve for \(s\):
\[
s = \sqrt[3]{0.125} = 0.5 \, \text{m}
\]
So, the dimensions of the cube are:
\[
\text{Length} = \text{Width} = \text{Height} = 0.5 \, \text{m}
\]
Thus, each side of the cube measures \(0.5 \, \text{m}\).