Answer :
Answer:
Step-by-step explanation:
To find the total volume of the aluminum alloy used in the hollow post, we need to consider the volume of the rectangular prism formed by the alloy, and then subtract the volume of the hollow space inside.
Given dimensions:
- Dimensions of the rectangular prism: 12 cm by 10 cm by 30 cm.
- Thickness of the aluminum alloy: 1.5 cm.
### Step 1: Calculate the volume of the solid rectangular prism
The volume \( V_{\text{solid}} \) of the solid rectangular prism is calculated using the formula for volume:
\[ V_{\text{solid}} = \text{length} \times \text{width} \times \text{height} \]
Substitute the given dimensions:
\[ V_{\text{solid}} = 12 \, \text{cm} \times 10 \, \text{cm} \times 30 \, \text{cm} \]
\[ V_{\text{solid}} = 3600 \, \text{cm}^3 \]
### Step 2: Calculate the volume of the hollow space
To find the volume of the hollow space inside the post, we need to subtract the volume of the inner rectangular prism from the volume of the outer rectangular prism.
Inner dimensions:
- Length: \( 12 \, \text{cm} - 2 \times 1.5 \, \text{cm} = 9 \, \text{cm} \)
- Width: \( 10 \, \text{cm} - 2 \times 1.5 \, \text{cm} = 7 \, \text{cm} \)
- Height: \( 30 \, \text{cm} - 2 \times 1.5 \, \text{cm} = 27 \, \text{cm} \)
Volume \( V_{\text{hollow}} \) of the hollow space:
\[ V_{\text{hollow}} = \text{length} \times \text{width} \times \text{height} \]
\[ V_{\text{hollow}} = 9 \, \text{cm} \times 7 \, \text{cm} \times 27 \, \text{cm} \]
\[ V_{\text{hollow}} = 1701 \, \text{cm}^3 \]
### Step 3: Find the total volume of alloy in the post
The total volume of the alloy in the post is the volume of the solid prism minus the volume of the hollow space:
\[ V_{\text{total}} = V_{\text{solid}} - V_{\text{hollow}} \]
\[ V_{\text{total}} = 3600 \, \text{cm}^3 - 1701 \, \text{cm}^3 \]
\[ V_{\text{total}} = 1899 \, \text{cm}^3 \]
### Conclusion
Therefore, the total volume of alloy in the post is \( \boxed{1899} \) cubic centimeters.
