To calculate the volume of the wood used to make the box, we first need to determine the external dimensions of the box (which include the thickness of the wood) and then the volume of the wood itself.
Step 1: Convert all dimensions to the same unit
We'll convert the internal dimensions from meters and centimeters to centimeters for consistency:
- Length: \(2 \text{ m} = 200 \text{ cm}\)
- Width: \(20 \text{ cm}\)
- Depth: \(22.5 \text{ cm}\)
Step 2: Determine the thickness of the wood
The thickness of the wood is given as \(2.5 \text{ cm}\).
Step 3: Calculate external dimensions
To find the external dimensions, we need to add twice the thickness of the wood (once for each side) to each internal dimension:
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External Length: \[ \text{External Length} = \text{Internal Length} + 2 \times \text{Thickness} \] \[ = 200 \text{ cm} + 2 \times 2.5 \text{ cm} = 200 \text{ cm} + 5 \text{ cm} = 205 \text{ cm} \]
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External Width: \[ \text{External Width} = \text{Internal Width} + 2 \times \text{Thickness} \] \[ = 20 \text{ cm} + 2 \times 2.5 \text{ cm} = 20 \text{ cm} + 5 \text{ cm} = 25 \text{ cm} \]
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External Depth: \[ \text{External Depth} = \text{Internal Depth} + 2 \times \text{Thickness} \] \[ = 22.5 \text{ cm} + 2 \times 2.5 \text{ cm} = 22.5 \text{ cm} + 5 \text{ cm} = 27.5 \text{ cm} \]
Step 4: Calculate the volume of the box
Using the internal and external dimensions, we can calculate the volumes.
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Internal Volume: \[ \text{Internal Volume} = \text{Length} \times \text{Width} \times \text{Depth} \] \[ = 200 \text{ cm} \times 20 \text{ cm} \times 22.5 \text{ cm} = 90000 \text{ cm}^3 \]
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External Volume: \[ \text{External Volume} = \text{External Length} \times \text{External Width} \times \text{External Depth} \] \[ = 205 \text{ cm} \times 25 \text{ cm} \times 27.5 \text{ cm} \] Calculating this step-by-step: \[ 205 \text{ cm} \times 25 \text{ cm} = 5125 \text{ cm}^2 \] \[ 5125 \text{ cm}^2 \times 27.5 \text{ cm} = 140968.75 \text{ cm}^3 \]
Step 5: Calculate the volume of the wood
The volume of the wood is given by the difference between the external volume and the internal volume: \[ \text{Volume of the wood} = \text{External Volume} - \text{Internal Volume} \] \[ = 140968.75 \text{ cm}^3 - 90000 \text{ cm}^3 = 50968.75 \text{ cm}^3 \]
Final Result
Thus, the volume of the wood used to construct the box is: \[ \text{Volume of the wood} \approx 50968.75 , \text{cm}^3 \]