To find the volume of the wood used to make the open box, we first need to calculate the volume of the external dimensions and then the internal dimensions.
Step 1: Calculate the external volume
The external dimensions of the box are:
- Length \( L_{\text{external}} = 51 , \text{cm} \)
- Width \( W_{\text{external}} = 45 , \text{cm} \)
- Height \( H_{\text{external}} = 21 , \text{cm} \)
The external volume \( V_{\text{external}} \) can be calculated using the formula for the volume of a cuboid:
\[ V_{\text{external}} = L_{\text{external}} \times W_{\text{external}} \times H_{\text{external}} \]
Plugging in the values:
\[ V_{\text{external}} = 51 \times 45 \times 21 \]
Calculating this:
\[ V_{\text{external}} = 51 \times 45 = 2295 , \text{cm}^2 \] \[ V_{\text{external}} = 2295 \times 21 = 48195 , \text{cm}^3 \]
Step 2: Calculate the internal volume
The thickness of the wood is 0.5 cm. We need to subtract twice the thickness from each dimension to find the internal dimensions.
- Length \( L_{\text{internal}} = L_{\text{external}} - 2 \times 0.5 = 51 - 1 = 50 , \text{cm} \)
- Width \( W_{\text{internal}} = W_{\text{external}} - 2 \times 0.5 = 45 - 1 = 44 , \text{cm} \)
- Height \( H_{\text{internal}} = H_{\text{external}} - 0.5 = 21 - 0.5 = 20.5 , \text{cm} \)
Now we calculate the internal volume \( V_{\text{internal}} \):
\[ V_{\text{internal}} = L_{\text{internal}} \times W_{\text{internal}} \times H_{\text{internal}} \]
Plugging in the values:
\[ V_{\text{internal}} = 50 \times 44 \times 20.5 \]
Calculating this:
\[ V_{\text{internal}} = 50 \times 44 = 2200 , \text{cm}^2 \] \[ V_{\text{internal}} = 2200 \times 20.5 = 45100 , \text{cm}^3 \]
Step 3: Calculate the volume of the wood
The volume of the wood \( V_{\text{wood}} \) is the difference between the external volume and the internal volume:
\[ V_{\text{wood}} = V_{\text{external}} - V_{\text{internal}} \] \[ V_{\text{wood}} = 48195 - 45100 = 3005 , \text{cm}^3 \]
Conclusion
The volume of the wood used to make the open box is 3005 cm³.
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