To calculate the density of the steel hex nut, follow these steps:
1. Volume of the Prism
We are given that the volume of the hexagonal prism (hex nut) is \(0.45 \, \text{cm}^3\).
2. Volume of the Cylinder (Hole)
The diameter of the cylinder (hole) is \(0.4 \, \text{cm}\).
- Radius of the cylinder: \(0.4 \, \text{cm} / 2 = 0.2 \, \text{cm}\).
- Volume of the cylinder \(V = \pi r^2h\), where \(h\) is the height (or depth) of the hex nut, assumed to be \(1 \, \text{cm}\).
Plugging in the values:
[tex]\[
V_{\text{cylinder}} = \pi \times (0.2 \, \text{cm})^2 \times 1 \, \text{cm}
\][/tex]
[tex]\[
V_{\text{cylinder}} = \pi \times 0.04 \, \text{cm}^2 \times 1 \, \text{cm}
\][/tex]
[tex]\[
V_{\text{cylinder}} \approx 0.126 \, \text{cm}^3 \approx 0.13 \, \text{cm}^3 \quad (\text{rounded to the nearest hundredth})
\][/tex]
So, the volume of the cylinder \(V = 0.13 \, \text{cm}^3\).
3. Volume of the Steel Hex Nut
To find the volume of the steel hex nut (excluding the hole), subtract the volume of the cylinder from the total volume of the prism:
[tex]\[
V_{\text{steel}} = V_{\text{prism}} - V_{\text{cylinder}}
\][/tex]
[tex]\[
V_{\text{steel}} = 0.45 \, \text{cm}^3 - 0.13 \, \text{cm}^3
\][/tex]
[tex]\[
V_{\text{steel}} = 0.32 \, \text{cm}^3
\][/tex]
So, the volume of the steel hex nut \(V = 0.32 \, \text{cm}^3\).
4. Calculate the Density of the Steel
Density is mass divided by volume:
[tex]\[
\text{Density} = \frac{\text{Mass}}{\text{Volume}}
\][/tex]
Given the mass of the hex nut is \(3.03 \, \text{grams}\):
[tex]\[
\text{Density} = \frac{3.03 \, \text{grams}}{0.32 \, \text{cm}^3}
\][/tex]
[tex]\[
\text{Density} \approx 9.46875 \, \text{grams/cm}^3
\][/tex]
Therefore, the density of the steel is approximately [tex]\(9.47 \, \text{grams/cm}^3\)[/tex] (rounded to the nearest hundredth).
