Answer :
Alright, let's carefully work through the problems presented:
### 1.1.1 Calculate the maximum number of Coca-Cola pallets that could be loaded on the second trailer of the truck.
#### Step-by-Step Solution:
1. Trailer Dimensions:
- Length [tex]\( L_{\text{trailer}} = 8.1 \, \text{m} \)[/tex]
- Width [tex]\( W_{\text{trailer}} = 2.45 \, \text{m} \)[/tex]
- Height [tex]\( H_{\text{trailer}} = 2.6 \, \text{m} \)[/tex]
2. Convert Bottle Dimensions to Meters:
- Radius of a bottle [tex]\( r = 52 \, \text{mm} \times 0.001 \, \left(\frac{\text{m}}{\text{mm}}\right) = 0.052 \, \text{m} \)[/tex]
- Height of a bottle [tex]\( h_{\text{bottle}} = 327 \, \text{mm} \times 13 \times 0.001 \left(\frac{\text{m}}{\text{mm}}\right) = 4.251 \, \text{m} \)[/tex]
3. Pallet Dimensions:
- Area covered by bottles on the pallet: [tex]\( \text{Area} = 8 \times 8 \times (\text{base area of a bottle}) \)[/tex]
- Base area of a bottle = [tex]\( \pi r^2 \)[/tex]
- [tex]\(\text{Base area of pallet} = (\text{Area of} \, 1 \, \text{bottle}) \times 64 = 64 \times \pi \times 0.052^2 \approx 0.54431 \, \text{m}^2\)[/tex]
- Assuming the pallet is rectangular:
- Pallet Length [tex]\( L_{\text{pallet}} = 2 \times r \times 8 = 2 \times 0.052 \times 8 = 0.832 \, \text{m} \)[/tex]
- Pallet Width [tex]\( W_{\text{pallet}} = L_{\text{pallet}} / 2 = 0.832 / 2 = 0.416 \, \text{m} \)[/tex]
- Pallet Height: [tex]\( H_{\text{pallet}} = 0.327 \, \text{m} \)[/tex] (since it's the bottle height)
4. Calculate Volumes:
- Volume of the trailer [tex]\( V_{\text{trailer}} = L_{\text{trailer}} \times W_{\text{trailer}} \times H_{\text{trailer}} = (8.1) \times (2.45) \times (2.6) \approx 51.597 \, \text{m}^3 \)[/tex]
- Volume of one pallet [tex]\( V_{\text{pallet}} = L_{\text{pallet}} \times W_{\text{pallet}} \times H_{\text{pallet}} \approx 0.832 \times 0.416 \times 0.327 \approx 0.1130 \, \text{m}^3 \)[/tex]
5. Maximum Number of Pallets in Trailer:
- [tex]\(\text{Max number of pallets} = \left\lfloor \frac{V_{\text{trailer}}}{V_{\text{pallet}}} \right\rfloor \approx \left\lfloor \frac{51.597}{0.1130} \right\rfloor = \left\lfloor 456.68 \right\rfloor = 2 \)[/tex]
So, the maximum number of Coca-Cola pallets that can be loaded on the second trailer is [tex]\( \boxed{2} \)[/tex].
### 1.1.2 Verifying Duan's Statement
#### Step-by-Step Solution:
1. Van Load Size:
- Van capacity: [tex]\( 1.5 \, \text{tons} = 1500 \, \text{kg} \)[/tex]
2. Pallet Mass Estimate:
- Assume [tex]\( 1 \, \text{kg} \approx 1 \, \text{litre} \)[/tex]
- Volume of a pallet: [tex]\( 0.1130 \, \text{m}^3 = 113 \, \text{litres} \)[/tex]
- Weight of a pallet [tex]\( = 113 \times 1 = 113 \, \text{kg} \)[/tex]
3. Maximum Number of Pallets the Van Can Carry:
- [tex]\(\text{Number of pallets in van} = \frac{1500 \, \text{kg}}{113 \, \text{kg/pallet}} \approx 13.27 \)[/tex]
Duan states that 12 pallets fit into the van, so:
4. Verification:
- To check if 12 pallets fit: [tex]\( 12 \leq 13.27 \)[/tex] which is true.
- Thus, Duan’s statement is correct as 12 pallets can fit into the van.
So, the statement by Duan is verified to be [tex]\( \boxed{\text{True}} \)[/tex].
### 1.1.1 Calculate the maximum number of Coca-Cola pallets that could be loaded on the second trailer of the truck.
#### Step-by-Step Solution:
1. Trailer Dimensions:
- Length [tex]\( L_{\text{trailer}} = 8.1 \, \text{m} \)[/tex]
- Width [tex]\( W_{\text{trailer}} = 2.45 \, \text{m} \)[/tex]
- Height [tex]\( H_{\text{trailer}} = 2.6 \, \text{m} \)[/tex]
2. Convert Bottle Dimensions to Meters:
- Radius of a bottle [tex]\( r = 52 \, \text{mm} \times 0.001 \, \left(\frac{\text{m}}{\text{mm}}\right) = 0.052 \, \text{m} \)[/tex]
- Height of a bottle [tex]\( h_{\text{bottle}} = 327 \, \text{mm} \times 13 \times 0.001 \left(\frac{\text{m}}{\text{mm}}\right) = 4.251 \, \text{m} \)[/tex]
3. Pallet Dimensions:
- Area covered by bottles on the pallet: [tex]\( \text{Area} = 8 \times 8 \times (\text{base area of a bottle}) \)[/tex]
- Base area of a bottle = [tex]\( \pi r^2 \)[/tex]
- [tex]\(\text{Base area of pallet} = (\text{Area of} \, 1 \, \text{bottle}) \times 64 = 64 \times \pi \times 0.052^2 \approx 0.54431 \, \text{m}^2\)[/tex]
- Assuming the pallet is rectangular:
- Pallet Length [tex]\( L_{\text{pallet}} = 2 \times r \times 8 = 2 \times 0.052 \times 8 = 0.832 \, \text{m} \)[/tex]
- Pallet Width [tex]\( W_{\text{pallet}} = L_{\text{pallet}} / 2 = 0.832 / 2 = 0.416 \, \text{m} \)[/tex]
- Pallet Height: [tex]\( H_{\text{pallet}} = 0.327 \, \text{m} \)[/tex] (since it's the bottle height)
4. Calculate Volumes:
- Volume of the trailer [tex]\( V_{\text{trailer}} = L_{\text{trailer}} \times W_{\text{trailer}} \times H_{\text{trailer}} = (8.1) \times (2.45) \times (2.6) \approx 51.597 \, \text{m}^3 \)[/tex]
- Volume of one pallet [tex]\( V_{\text{pallet}} = L_{\text{pallet}} \times W_{\text{pallet}} \times H_{\text{pallet}} \approx 0.832 \times 0.416 \times 0.327 \approx 0.1130 \, \text{m}^3 \)[/tex]
5. Maximum Number of Pallets in Trailer:
- [tex]\(\text{Max number of pallets} = \left\lfloor \frac{V_{\text{trailer}}}{V_{\text{pallet}}} \right\rfloor \approx \left\lfloor \frac{51.597}{0.1130} \right\rfloor = \left\lfloor 456.68 \right\rfloor = 2 \)[/tex]
So, the maximum number of Coca-Cola pallets that can be loaded on the second trailer is [tex]\( \boxed{2} \)[/tex].
### 1.1.2 Verifying Duan's Statement
#### Step-by-Step Solution:
1. Van Load Size:
- Van capacity: [tex]\( 1.5 \, \text{tons} = 1500 \, \text{kg} \)[/tex]
2. Pallet Mass Estimate:
- Assume [tex]\( 1 \, \text{kg} \approx 1 \, \text{litre} \)[/tex]
- Volume of a pallet: [tex]\( 0.1130 \, \text{m}^3 = 113 \, \text{litres} \)[/tex]
- Weight of a pallet [tex]\( = 113 \times 1 = 113 \, \text{kg} \)[/tex]
3. Maximum Number of Pallets the Van Can Carry:
- [tex]\(\text{Number of pallets in van} = \frac{1500 \, \text{kg}}{113 \, \text{kg/pallet}} \approx 13.27 \)[/tex]
Duan states that 12 pallets fit into the van, so:
4. Verification:
- To check if 12 pallets fit: [tex]\( 12 \leq 13.27 \)[/tex] which is true.
- Thus, Duan’s statement is correct as 12 pallets can fit into the van.
So, the statement by Duan is verified to be [tex]\( \boxed{\text{True}} \)[/tex].
