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?
a. 50 m
b. 0.5 m
c. 5 m
d. 0.05 m
a. 50 m
b. 0.5 m
c. 5 m
d. 0.05 m
Answers
GPT-4o mini
To find the dimensions of the cube, we can use the relationship between mass, density, and volume. The formula is:
\[
\text{Density} (\rho) = \frac{\text{Mass} (m)}{\text{Volume} (V)}
\]
We can rearrange this formula to find the volume:
\[
V = \frac{m}{\rho}
\]
Given:
- Mass \( m = 1000 \, \text{kg} \)
- Density \( \rho = 8000 \, \text{kg/m}^3 \)
Now, substituting the values into the volume formula:
\[
V = \frac{1000 \, \text{kg}}{8000 \, \text{kg/m}^3} = \frac{1000}{8000} \, \text{m}^3 = 0.125 \, \text{m}^3
\]
Next, for a cube, the volume \( V \) is given by the formula:
\[
V = a^3
\]
where \( a \) is the length of a side of the cube. We can now find \( a \) by solving:
\[
a^3 = 0.125 \, \text{m}^3
\]
Taking the cube root of both sides:
\[
a = \sqrt[3]{0.125} = 0.5 \, \text{m}
\]
Thus, the dimensions of the cube are:
**b. 0.5 m**
\[
\text{Density} (\rho) = \frac{\text{Mass} (m)}{\text{Volume} (V)}
\]
We can rearrange this formula to find the volume:
\[
V = \frac{m}{\rho}
\]
Given:
- Mass \( m = 1000 \, \text{kg} \)
- Density \( \rho = 8000 \, \text{kg/m}^3 \)
Now, substituting the values into the volume formula:
\[
V = \frac{1000 \, \text{kg}}{8000 \, \text{kg/m}^3} = \frac{1000}{8000} \, \text{m}^3 = 0.125 \, \text{m}^3
\]
Next, for a cube, the volume \( V \) is given by the formula:
\[
V = a^3
\]
where \( a \) is the length of a side of the cube. We can now find \( a \) by solving:
\[
a^3 = 0.125 \, \text{m}^3
\]
Taking the cube root of both sides:
\[
a = \sqrt[3]{0.125} = 0.5 \, \text{m}
\]
Thus, the dimensions of the cube are:
**b. 0.5 m**