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/m3 ?
Answers
GPT-4o mini
To find the dimensions of the cube, we can use the relationship between mass, density, and volume. The formula for density is:
\[
\text{Density} = \frac{\text{Mass}}{\text{Volume}}
\]
From this, we can express volume as:
\[
\text{Volume} = \frac{\text{Mass}}{\text{Density}}
\]
Given:
- Mass = 1,000 kg
- Density of steel = 8,000 kg/m³
We can calculate the volume of the cube:
\[
\text{Volume} = \frac{1,000 \, \text{kg}}{8,000 \, \text{kg/m}^3} = \frac{1,000}{8,000} \, \text{m}^3 = 0.125 \, \text{m}^3
\]
For a cube, the volume \( V \) is also given by the formula:
\[
V = s^3
\]
where \( s \) is the length of a side of the cube.
Now we can set the two expressions for volume equal to each other:
\[
s^3 = 0.125 \, \text{m}^3
\]
To find \( s \), we take the cube root of both sides:
\[
s = \sqrt[3]{0.125}
\]
Calculating the cube root:
\[
s = 0.5 \, \text{m}
\]
Thus, the dimensions of the cube are \( 0.5 \, \text{m} \times 0.5 \, \text{m} \times 0.5 \, \text{m} \).
\[
\text{Density} = \frac{\text{Mass}}{\text{Volume}}
\]
From this, we can express volume as:
\[
\text{Volume} = \frac{\text{Mass}}{\text{Density}}
\]
Given:
- Mass = 1,000 kg
- Density of steel = 8,000 kg/m³
We can calculate the volume of the cube:
\[
\text{Volume} = \frac{1,000 \, \text{kg}}{8,000 \, \text{kg/m}^3} = \frac{1,000}{8,000} \, \text{m}^3 = 0.125 \, \text{m}^3
\]
For a cube, the volume \( V \) is also given by the formula:
\[
V = s^3
\]
where \( s \) is the length of a side of the cube.
Now we can set the two expressions for volume equal to each other:
\[
s^3 = 0.125 \, \text{m}^3
\]
To find \( s \), we take the cube root of both sides:
\[
s = \sqrt[3]{0.125}
\]
Calculating the cube root:
\[
s = 0.5 \, \text{m}
\]
Thus, the dimensions of the cube are \( 0.5 \, \text{m} \times 0.5 \, \text{m} \times 0.5 \, \text{m} \).