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Sagot :
Let's solve each step in detail:
1. Calculate the volume of the prism.
The volume of a hexagonal prism can be determined by first calculating the area of one of its hexagonal faces and then multiplying by the height of the prism. The area [tex]\( A \)[/tex] of a hexagon can be found using the formula:
[tex]\[ A = \frac{1}{2} \times \text{apothem} \times \text{perimeter} \][/tex]
Given:
- Apothem ([tex]\( a \)[/tex]) = 1 cm
- Perimeter ([tex]\( P \)[/tex]) = 2 cm
[tex]\[ A = \frac{1}{2} \times 1 \text{ cm} \times 2 \text{ cm} = 1 \text{ cm}^2 \][/tex]
Next, we multiply the area of the hexagon by the height ([tex]\( h \)[/tex]) of the prism to get the volume:
[tex]\[ \text{Volume of the prism} = A \times h \][/tex]
Given:
- Height ([tex]\( h \)[/tex]) = 2 cm
[tex]\[ \text{Volume of the prism} = 1 \text{ cm}^2 \times 2 \text{ cm} = 2 \text{ cm}^3 \][/tex]
So, [tex]\( V = 2 \text{ cm}^3 \)[/tex].
2. Calculate the volume of the cylinder.
To find the volume of the cylindrical hole, we use the formula for the volume of a cylinder:
[tex]\[ V = \pi \times r^2 \times h \][/tex]
Given:
- Diameter of the hole = 0.4 cm
- Radius ([tex]\( r \)[/tex]) = Diameter / 2 = 0.4 cm / 2 = 0.2 cm
- Height ([tex]\( h \)[/tex]) = 2 cm
[tex]\[ V = \pi \times (0.2 \text{ cm})^2 \times 2 \text{ cm} \][/tex]
[tex]\[ V \approx \pi \times 0.04 \text{ cm}^2 \times 2 \text{ cm} \][/tex]
[tex]\[ V \approx 0.25132741228718347 \text{ cm}^3 \][/tex]
So, [tex]\( V \approx 0.251 \text{ cm}^3 \)[/tex] (to the nearest hundredth).
3. Find the volume of the composite figure.
To find the total volume of the composite figure, subtract the volume of the cylindrical hole from the volume of the hexagonal prism:
[tex]\[ \text{Total volume} = \text{Volume of the prism} - \text{Volume of the cylinder} \][/tex]
[tex]\[ \text{Total volume} = 2 \text{ cm}^3 - 0.25132741228718347 \text{ cm}^3 \][/tex]
[tex]\[ \text{Total volume} \approx 1.749 \text{ cm}^3 \][/tex]
So, [tex]\( V = 1.749 \text{ cm}^3 \)[/tex].
4. Calculate the density by dividing the mass by the volume.
Density ([tex]\( \rho \)[/tex]) is given by:
[tex]\[ \rho = \frac{\text{mass}}{\text{volume}} \][/tex]
Given:
- Mass = 3.03 grams
- Volume = 1.749 cm³
[tex]\[ \rho = \frac{3.03 \text{ g}}{1.749 \text{ cm}^3} \][/tex]
[tex]\[ \rho \approx 1.733 \text{ g/cm}^3 \][/tex]
So, the density [tex]\( d = 1.733 \text{ g/cm}^3 \)[/tex].
Thus, the detailed steps yield:
1. [tex]\( V_{prism} = 2 \text{ cm}^3 \)[/tex]
2. [tex]\( V_{cylinder} \approx 0.251 \text{ cm}^3 \)[/tex]
3. [tex]\( V_{composite} \approx 1.749 \text{ cm}^3 \)[/tex]
4. [tex]\( d \approx 1.733 \text{ g/cm}^3 \)[/tex]
1. Calculate the volume of the prism.
The volume of a hexagonal prism can be determined by first calculating the area of one of its hexagonal faces and then multiplying by the height of the prism. The area [tex]\( A \)[/tex] of a hexagon can be found using the formula:
[tex]\[ A = \frac{1}{2} \times \text{apothem} \times \text{perimeter} \][/tex]
Given:
- Apothem ([tex]\( a \)[/tex]) = 1 cm
- Perimeter ([tex]\( P \)[/tex]) = 2 cm
[tex]\[ A = \frac{1}{2} \times 1 \text{ cm} \times 2 \text{ cm} = 1 \text{ cm}^2 \][/tex]
Next, we multiply the area of the hexagon by the height ([tex]\( h \)[/tex]) of the prism to get the volume:
[tex]\[ \text{Volume of the prism} = A \times h \][/tex]
Given:
- Height ([tex]\( h \)[/tex]) = 2 cm
[tex]\[ \text{Volume of the prism} = 1 \text{ cm}^2 \times 2 \text{ cm} = 2 \text{ cm}^3 \][/tex]
So, [tex]\( V = 2 \text{ cm}^3 \)[/tex].
2. Calculate the volume of the cylinder.
To find the volume of the cylindrical hole, we use the formula for the volume of a cylinder:
[tex]\[ V = \pi \times r^2 \times h \][/tex]
Given:
- Diameter of the hole = 0.4 cm
- Radius ([tex]\( r \)[/tex]) = Diameter / 2 = 0.4 cm / 2 = 0.2 cm
- Height ([tex]\( h \)[/tex]) = 2 cm
[tex]\[ V = \pi \times (0.2 \text{ cm})^2 \times 2 \text{ cm} \][/tex]
[tex]\[ V \approx \pi \times 0.04 \text{ cm}^2 \times 2 \text{ cm} \][/tex]
[tex]\[ V \approx 0.25132741228718347 \text{ cm}^3 \][/tex]
So, [tex]\( V \approx 0.251 \text{ cm}^3 \)[/tex] (to the nearest hundredth).
3. Find the volume of the composite figure.
To find the total volume of the composite figure, subtract the volume of the cylindrical hole from the volume of the hexagonal prism:
[tex]\[ \text{Total volume} = \text{Volume of the prism} - \text{Volume of the cylinder} \][/tex]
[tex]\[ \text{Total volume} = 2 \text{ cm}^3 - 0.25132741228718347 \text{ cm}^3 \][/tex]
[tex]\[ \text{Total volume} \approx 1.749 \text{ cm}^3 \][/tex]
So, [tex]\( V = 1.749 \text{ cm}^3 \)[/tex].
4. Calculate the density by dividing the mass by the volume.
Density ([tex]\( \rho \)[/tex]) is given by:
[tex]\[ \rho = \frac{\text{mass}}{\text{volume}} \][/tex]
Given:
- Mass = 3.03 grams
- Volume = 1.749 cm³
[tex]\[ \rho = \frac{3.03 \text{ g}}{1.749 \text{ cm}^3} \][/tex]
[tex]\[ \rho \approx 1.733 \text{ g/cm}^3 \][/tex]
So, the density [tex]\( d = 1.733 \text{ g/cm}^3 \)[/tex].
Thus, the detailed steps yield:
1. [tex]\( V_{prism} = 2 \text{ cm}^3 \)[/tex]
2. [tex]\( V_{cylinder} \approx 0.251 \text{ cm}^3 \)[/tex]
3. [tex]\( V_{composite} \approx 1.749 \text{ cm}^3 \)[/tex]
4. [tex]\( d \approx 1.733 \text{ g/cm}^3 \)[/tex]
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