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### Step-by-Step Solution
#### Step 1: Calculate the Volume of the Cylindrical Tank
1. Diameter of the cylindrical tank: [tex]\(2.0 \, \text{cm}\)[/tex]
2. Radius of the cylindrical tank: [tex]\( \frac{2.0}{2} = 1.0 \, \text{cm} \)[/tex]
3. Height of the cylindrical tank: [tex]\(3.0 \, \text{cm}\)[/tex]
The formula for the volume [tex]\(V\)[/tex] of a cylinder is given by:
[tex]\[ V = \pi r^2 h \][/tex]
Substitute the values:
[tex]\[ V = \pi \times (1.0)^2 \times 3.0 \][/tex]
[tex]\[ V = \pi \times 1.0^2 \times 3.0 \][/tex]
[tex]\[ V = 3.0\pi \][/tex]
Using this value, we find the volume of the cylinder:
[tex]\[ V_{\text{cylinder}} = 9.42477796076938 \, \text{cm}^3 \][/tex]
#### Step 2: Determine the Volume of the Frustum of a Cone
For the frustum of a cone, we are given the following dimensions:
1. Diameter of the top end of the frustum: [tex]\(1.0 \, \text{cm} \)[/tex]
2. Radius of the top end of the frustum: [tex]\( \frac{1.0}{2} = 0.5 \, \text{cm} \)[/tex]
3. Diameter of the bottom end of the frustum: [tex]\( 2.0 \, \text{cm} \)[/tex]
4. Radius of the bottom end of the frustum: [tex]\( \frac{2.0}{2} = 1.0 \, \text{cm} \)[/tex]
5. Height of the frustum: [tex]\( h \, \text{cm} \)[/tex]
The formula for the volume [tex]\(V\)[/tex] of a frustum of a cone is given by:
[tex]\[ V = \frac{1}{3} \pi h (R^2 + Rr + r^2) \][/tex]
Where:
- [tex]\(R\)[/tex] is the radius of the bottom end ([tex]\(1.0 \, \text{cm}\)[/tex])
- [tex]\(r\)[/tex] is the radius of the top end ([tex]\(0.5 \, \text{cm}\)[/tex])
Substitute these values into the formula:
[tex]\[ V_{\text{frustum}} = \frac{1}{3} \pi h \left(1.0^2 + 1.0 \times 0.5 + 0.5^2\right) \][/tex]
[tex]\[ V_{\text{frustum}} = \frac{1}{3} \pi h (1.0 + 0.5 + 0.25) \][/tex]
[tex]\[ V_{\text{frustum}} = \frac{1}{3} \pi h (1.75) \][/tex]
[tex]\[ V_{\text{frustum}} = \frac{1.75}{3} \pi h \][/tex]
[tex]\[ V_{\text{frustum}} = \frac{7}{12} \pi h \][/tex]
Using this value, we find the volume in terms of [tex]\(h\)[/tex]:
[tex]\[ V_{\text{frustum}} = 1.83259571459405 h \, \text{cm}^3 \][/tex]
#### Step 3: Equate the Volume of the Cylinder to the Volume of the Frustum
To find the required height [tex]\(h\)[/tex] of the frustum, we set the volume of the cylinder equal to the volume of the frustum:
[tex]\[ 9.42477796076938 = 1.83259571459405 h \][/tex]
Solve for [tex]\(h\)[/tex]:
[tex]\[ h = \frac{9.42477796076938}{1.83259571459405} \][/tex]
[tex]\[ h = 5.14285714285713 \, \text{cm} \][/tex]
### Conclusion
Hence, the vertical height required for the frustum to replace the cylindrical tank with the same capacity is approximately [tex]\( 5.14 \, \text{cm} \)[/tex].
### Step-by-Step Solution
#### Step 1: Calculate the Volume of the Cylindrical Tank
1. Diameter of the cylindrical tank: [tex]\(2.0 \, \text{cm}\)[/tex]
2. Radius of the cylindrical tank: [tex]\( \frac{2.0}{2} = 1.0 \, \text{cm} \)[/tex]
3. Height of the cylindrical tank: [tex]\(3.0 \, \text{cm}\)[/tex]
The formula for the volume [tex]\(V\)[/tex] of a cylinder is given by:
[tex]\[ V = \pi r^2 h \][/tex]
Substitute the values:
[tex]\[ V = \pi \times (1.0)^2 \times 3.0 \][/tex]
[tex]\[ V = \pi \times 1.0^2 \times 3.0 \][/tex]
[tex]\[ V = 3.0\pi \][/tex]
Using this value, we find the volume of the cylinder:
[tex]\[ V_{\text{cylinder}} = 9.42477796076938 \, \text{cm}^3 \][/tex]
#### Step 2: Determine the Volume of the Frustum of a Cone
For the frustum of a cone, we are given the following dimensions:
1. Diameter of the top end of the frustum: [tex]\(1.0 \, \text{cm} \)[/tex]
2. Radius of the top end of the frustum: [tex]\( \frac{1.0}{2} = 0.5 \, \text{cm} \)[/tex]
3. Diameter of the bottom end of the frustum: [tex]\( 2.0 \, \text{cm} \)[/tex]
4. Radius of the bottom end of the frustum: [tex]\( \frac{2.0}{2} = 1.0 \, \text{cm} \)[/tex]
5. Height of the frustum: [tex]\( h \, \text{cm} \)[/tex]
The formula for the volume [tex]\(V\)[/tex] of a frustum of a cone is given by:
[tex]\[ V = \frac{1}{3} \pi h (R^2 + Rr + r^2) \][/tex]
Where:
- [tex]\(R\)[/tex] is the radius of the bottom end ([tex]\(1.0 \, \text{cm}\)[/tex])
- [tex]\(r\)[/tex] is the radius of the top end ([tex]\(0.5 \, \text{cm}\)[/tex])
Substitute these values into the formula:
[tex]\[ V_{\text{frustum}} = \frac{1}{3} \pi h \left(1.0^2 + 1.0 \times 0.5 + 0.5^2\right) \][/tex]
[tex]\[ V_{\text{frustum}} = \frac{1}{3} \pi h (1.0 + 0.5 + 0.25) \][/tex]
[tex]\[ V_{\text{frustum}} = \frac{1}{3} \pi h (1.75) \][/tex]
[tex]\[ V_{\text{frustum}} = \frac{1.75}{3} \pi h \][/tex]
[tex]\[ V_{\text{frustum}} = \frac{7}{12} \pi h \][/tex]
Using this value, we find the volume in terms of [tex]\(h\)[/tex]:
[tex]\[ V_{\text{frustum}} = 1.83259571459405 h \, \text{cm}^3 \][/tex]
#### Step 3: Equate the Volume of the Cylinder to the Volume of the Frustum
To find the required height [tex]\(h\)[/tex] of the frustum, we set the volume of the cylinder equal to the volume of the frustum:
[tex]\[ 9.42477796076938 = 1.83259571459405 h \][/tex]
Solve for [tex]\(h\)[/tex]:
[tex]\[ h = \frac{9.42477796076938}{1.83259571459405} \][/tex]
[tex]\[ h = 5.14285714285713 \, \text{cm} \][/tex]
### Conclusion
Hence, the vertical height required for the frustum to replace the cylindrical tank with the same capacity is approximately [tex]\( 5.14 \, \text{cm} \)[/tex].
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