Answer :
Let's break down the problem step by step:
1. Determine the Radii:
- Given the external diameter is 16 cm.
- Therefore, the external radius [tex]\( R \)[/tex] is:
[tex]\[ R = \frac{16}{2} = 8 \text{ cm} \][/tex]
- The uniform thickness of the mould is 0.8 cm.
- Hence, the internal radius [tex]\( r \)[/tex] is:
[tex]\[ r = 8 \text{ cm} - 0.8 \text{ cm} = 7.2 \text{ cm} \][/tex]
2. Calculate the Volume of the External Sphere:
- The formula for the volume of a sphere is:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
- For the external sphere with radius [tex]\( R = 8 \text{ cm} \)[/tex]:
[tex]\[ V_{\text{external}} = \frac{4}{3} \pi (8)^3 = \frac{4}{3} \pi (512) = \frac{2048}{3} \pi \text{ cm}^3 \][/tex]
3. Calculate the Volume of the Internal Sphere:
- For the internal sphere with radius [tex]\( r = 7.2 \text{ cm} \)[/tex]:
[tex]\[ V_{\text{internal}} = \frac{4}{3} \pi (7.2)^3 \][/tex]
- Let's compute [tex]\( (7.2)^3 \)[/tex]:
[tex]\[ (7.2)^3 = 7.2 \times 7.2 \times 7.2 = 373.248 \][/tex]
- Thus,
[tex]\[ V_{\text{internal}} = \frac{4}{3} \pi 373.248 = \frac{1492.992}{3} \pi \text{ cm}^3 \][/tex]
4. Calculate the Volume of Steel for One Mould:
- The volume of steel is the difference between the external and internal volumes:
[tex]\[ V_{\text{steel}} = V_{\text{external}} - V_{\text{internal}} \][/tex]
[tex]\[ V_{\text{steel}} = \frac{2048}{3} \pi \text{ cm}^3 - \frac{1492.992}{3} \pi \text{ cm}^3 \][/tex]
[tex]\[ V_{\text{steel}} = \frac{2048 - 1492.992}{3} \pi \][/tex]
[tex]\[ V_{\text{steel}} = \frac{555.008}{3} \pi \][/tex]
[tex]\[ V_{\text{steel}} = 185.0027 \pi \text{ cm}^3 \][/tex]
5. Calculate the Volume for 100 Moulds:
- For 100 moulds, the total volume of steel required is:
[tex]\[ V_{\text{steel, total}} = 100 \times 185.0027 \pi = 18500.27 \pi \text{ cm}^3 \][/tex]
6. Approximate the Value:
- Using [tex]\( \pi \approx 3.14159 \)[/tex]:
[tex]\[ V_{\text{steel, total}} \approx 18500.27 \times 3.14159 \approx 58107.89 \text{ cm}^3 \][/tex]
Therefore, the volume of steel required to make 100 such hollow spherical moulds is approximately [tex]\( 58107.89 \text{ cm}^3 \)[/tex].
1. Determine the Radii:
- Given the external diameter is 16 cm.
- Therefore, the external radius [tex]\( R \)[/tex] is:
[tex]\[ R = \frac{16}{2} = 8 \text{ cm} \][/tex]
- The uniform thickness of the mould is 0.8 cm.
- Hence, the internal radius [tex]\( r \)[/tex] is:
[tex]\[ r = 8 \text{ cm} - 0.8 \text{ cm} = 7.2 \text{ cm} \][/tex]
2. Calculate the Volume of the External Sphere:
- The formula for the volume of a sphere is:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
- For the external sphere with radius [tex]\( R = 8 \text{ cm} \)[/tex]:
[tex]\[ V_{\text{external}} = \frac{4}{3} \pi (8)^3 = \frac{4}{3} \pi (512) = \frac{2048}{3} \pi \text{ cm}^3 \][/tex]
3. Calculate the Volume of the Internal Sphere:
- For the internal sphere with radius [tex]\( r = 7.2 \text{ cm} \)[/tex]:
[tex]\[ V_{\text{internal}} = \frac{4}{3} \pi (7.2)^3 \][/tex]
- Let's compute [tex]\( (7.2)^3 \)[/tex]:
[tex]\[ (7.2)^3 = 7.2 \times 7.2 \times 7.2 = 373.248 \][/tex]
- Thus,
[tex]\[ V_{\text{internal}} = \frac{4}{3} \pi 373.248 = \frac{1492.992}{3} \pi \text{ cm}^3 \][/tex]
4. Calculate the Volume of Steel for One Mould:
- The volume of steel is the difference between the external and internal volumes:
[tex]\[ V_{\text{steel}} = V_{\text{external}} - V_{\text{internal}} \][/tex]
[tex]\[ V_{\text{steel}} = \frac{2048}{3} \pi \text{ cm}^3 - \frac{1492.992}{3} \pi \text{ cm}^3 \][/tex]
[tex]\[ V_{\text{steel}} = \frac{2048 - 1492.992}{3} \pi \][/tex]
[tex]\[ V_{\text{steel}} = \frac{555.008}{3} \pi \][/tex]
[tex]\[ V_{\text{steel}} = 185.0027 \pi \text{ cm}^3 \][/tex]
5. Calculate the Volume for 100 Moulds:
- For 100 moulds, the total volume of steel required is:
[tex]\[ V_{\text{steel, total}} = 100 \times 185.0027 \pi = 18500.27 \pi \text{ cm}^3 \][/tex]
6. Approximate the Value:
- Using [tex]\( \pi \approx 3.14159 \)[/tex]:
[tex]\[ V_{\text{steel, total}} \approx 18500.27 \times 3.14159 \approx 58107.89 \text{ cm}^3 \][/tex]
Therefore, the volume of steel required to make 100 such hollow spherical moulds is approximately [tex]\( 58107.89 \text{ cm}^3 \)[/tex].
