To determine the volume of wood used in the closed box, we will first calculate the volume of the outer dimensions of the box, and then we will calculate the volume of the inner dimensions, which is the volume of the empty space inside the box. The volume of wood used will be the difference between these two volumes.
Step 1: Calculate the outer volume
The outer dimensions of the box are given as:
- Length (L) = 42 cm
- Width (W) = 30 cm
- Height (H) = 20 cm
The volume of the outer box (V_outer) is calculated using the formula:
\[ V_{\text{outer}} = L \times W \times H \] \[ V_{\text{outer}} = 42 , \text{cm} \times 30 , \text{cm} \times 20 , \text{cm} \] \[ V_{\text{outer}} = 25200 , \text{cm}^3 \]
Step 2: Calculate the inner volume
Since the box is made of wood with a thickness of 1 cm, we need to subtract this thickness from all sides to get the inner dimensions:
- Inner Length (L_inner) = Outer Length - 2 × Thickness = 42 cm - 2 × 1 cm = 40 cm
- Inner Width (W_inner) = Outer Width - 2 × Thickness = 30 cm - 2 × 1 cm = 28 cm
- Inner Height (H_inner) = Outer Height - 2 × Thickness = 20 cm - 2 × 1 cm = 18 cm
Now we can calculate the volume of the inner box (V_inner):
\[ V_{\text{inner}} = L_{\text{inner}} \times W_{\text{inner}} \times H_{\text{inner}} \] \[ V_{\text{inner}} = 40 , \text{cm} \times 28 , \text{cm} \times 18 , \text{cm} \] \[ V_{\text{inner}} = 20160 , \text{cm}^3 \]
Step 3: Calculate the volume of the wood
The volume of the wood used in the box is the difference between the outer volume and the inner volume:
\[ V_{\text{wood}} = V_{\text{outer}} - V_{\text{inner}} \] \[ V_{\text{wood}} = 25200 , \text{cm}^3 - 20160 , \text{cm}^3 \] \[ V_{\text{wood}} = 5040 , \text{cm}^3 \]
Thus, the volume of wood used in the box is \( \boxed{5040} , \text{cm}^3 \).