Answer :
To determine which locomotive is needed to haul each container, we need to calculate the volume of oil in each cylindrical container and match this volume to the appropriate locomotive from the provided capacity table. We follow these steps:
### Cylinder A:
1. Dimensions: Length = 40 ft., Diameter = 12 ft.
2. Fill Level: half (0.5)
The volume of a cylinder is given by the formula:
[tex]\[ V = \pi r^2 h \][/tex]
Where [tex]\( r \)[/tex] is the radius and [tex]\( h \)[/tex] is the height (length).
- Calculate the radius:
[tex]\[ r = \frac{\text{Diameter}}{2} = \frac{12}{2} = 6 \, \text{ft} \][/tex]
- Calculate the volume:
[tex]\[ V = \pi (6)^2 (40) = \pi (36) (40) = 1440 \pi \][/tex]
- Since it is filled to half its capacity:
[tex]\[ V_{\text{fill}} = 1440 \pi \times 0.5 = 720 \pi \approx 2261.95 \, \text{cubic feet} \][/tex]
According to the capacity table, 2261.95 cubic feet falls in the range for locomotive BR73 (1,500 - 2,500 cubic feet).
### Cylinder B:
1. Dimensions: Length = 24 ft., Diameter = 8 ft.
2. Fill Level: full (1.0)
- Calculate the radius:
[tex]\[ r = \frac{\text{Diameter}}{2} = \frac{8}{2} = 4 \, \text{ft} \][/tex]
- Calculate the volume:
[tex]\[ V = \pi (4)^2 (24) = \pi (16) (24) = 384 \pi \approx 1206.37 \, \text{cubic feet} \][/tex]
According to the capacity table, 1206.37 cubic feet falls in the range for locomotive CG35 (750 - 1,500 cubic feet).
### Cylinder C:
1. Dimensions: Length = 16 ft., Diameter = 16 ft.
2. Fill Level: full (1.0)
- Calculate the radius:
[tex]\[ r = \frac{\text{Diameter}}{2} = \frac{16}{2} = 8 \, \text{ft} \][/tex]
- Calculate the volume:
[tex]\[ V = \pi (8)^2 (16) = \pi (64) (16) = 1024 \pi \approx 3216.99 \, \text{cubic feet} \][/tex]
According to the capacity table, 3216.99 cubic feet falls in the range for locomotive YH61 (2,500 - 3,500 cubic feet).
### Cylinder D:
1. Dimensions: Length = 6 ft., Diameter = 12 ft.
2. Fill Level: full (1.0)
- Calculate the radius:
[tex]\[ r = \frac{\text{Diameter}}{2} = \frac{12}{2} = 6 \, \text{ft} \][/tex]
- Calculate the volume:
[tex]\[ V = \pi (6)^2 (6) = \pi (36) (6) = 216 \pi \approx 678.58 \, \text{cubic feet} \][/tex]
According to the capacity table, 678.58 cubic feet falls in the range for locomotive A450 (0 - 750 cubic feet).
### Summary:
- Cylinder A: BR73
- Cylinder B: CG35
- Cylinder C: YH61
- Cylinder D: A450
These are the locomotives needed to haul each cylindrical container based on their volumes.
### Cylinder A:
1. Dimensions: Length = 40 ft., Diameter = 12 ft.
2. Fill Level: half (0.5)
The volume of a cylinder is given by the formula:
[tex]\[ V = \pi r^2 h \][/tex]
Where [tex]\( r \)[/tex] is the radius and [tex]\( h \)[/tex] is the height (length).
- Calculate the radius:
[tex]\[ r = \frac{\text{Diameter}}{2} = \frac{12}{2} = 6 \, \text{ft} \][/tex]
- Calculate the volume:
[tex]\[ V = \pi (6)^2 (40) = \pi (36) (40) = 1440 \pi \][/tex]
- Since it is filled to half its capacity:
[tex]\[ V_{\text{fill}} = 1440 \pi \times 0.5 = 720 \pi \approx 2261.95 \, \text{cubic feet} \][/tex]
According to the capacity table, 2261.95 cubic feet falls in the range for locomotive BR73 (1,500 - 2,500 cubic feet).
### Cylinder B:
1. Dimensions: Length = 24 ft., Diameter = 8 ft.
2. Fill Level: full (1.0)
- Calculate the radius:
[tex]\[ r = \frac{\text{Diameter}}{2} = \frac{8}{2} = 4 \, \text{ft} \][/tex]
- Calculate the volume:
[tex]\[ V = \pi (4)^2 (24) = \pi (16) (24) = 384 \pi \approx 1206.37 \, \text{cubic feet} \][/tex]
According to the capacity table, 1206.37 cubic feet falls in the range for locomotive CG35 (750 - 1,500 cubic feet).
### Cylinder C:
1. Dimensions: Length = 16 ft., Diameter = 16 ft.
2. Fill Level: full (1.0)
- Calculate the radius:
[tex]\[ r = \frac{\text{Diameter}}{2} = \frac{16}{2} = 8 \, \text{ft} \][/tex]
- Calculate the volume:
[tex]\[ V = \pi (8)^2 (16) = \pi (64) (16) = 1024 \pi \approx 3216.99 \, \text{cubic feet} \][/tex]
According to the capacity table, 3216.99 cubic feet falls in the range for locomotive YH61 (2,500 - 3,500 cubic feet).
### Cylinder D:
1. Dimensions: Length = 6 ft., Diameter = 12 ft.
2. Fill Level: full (1.0)
- Calculate the radius:
[tex]\[ r = \frac{\text{Diameter}}{2} = \frac{12}{2} = 6 \, \text{ft} \][/tex]
- Calculate the volume:
[tex]\[ V = \pi (6)^2 (6) = \pi (36) (6) = 216 \pi \approx 678.58 \, \text{cubic feet} \][/tex]
According to the capacity table, 678.58 cubic feet falls in the range for locomotive A450 (0 - 750 cubic feet).
### Summary:
- Cylinder A: BR73
- Cylinder B: CG35
- Cylinder C: YH61
- Cylinder D: A450
These are the locomotives needed to haul each cylindrical container based on their volumes.
