To calculate the area of the wood used in the open cuboidal box, we first need to determine the external and internal dimensions of the box. The box is 0.5 cm thick, so we will subtract this thickness from each dimension to find the internal dimensions.
Step 1: Calculate the Internal Dimensions
External Dimensions:
- Length = 52 cm
- Width = 45 cm
- Depth = 21 cm
Thickness of the wood = 0.5 cm
Internal Dimensions:
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Internal Length = External Length - 2 × Thickness \[ 52 , \text{cm} - 2 \times 0.5 , \text{cm} = 52 , \text{cm} - 1 , \text{cm} = 51 , \text{cm} \]
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Internal Width = External Width - 2 × Thickness \[ 45 , \text{cm} - 2 \times 0.5 , \text{cm} = 45 , \text{cm} - 1 , \text{cm} = 44 , \text{cm} \]
-
Internal Depth = External Depth - Thickness (only on the bottom) \[ 21 , \text{cm} - 0.5 , \text{cm} = 20.5 , \text{cm} \]
Step 2: Calculate the Areas
External Area Calculation: The external surface area of the open box (which has no top) can be calculated using the formula for the surface area of a cuboid: \[ \text{External Surface Area} = \text{Length} \times \text{Width} + \text{Length} \times \text{Depth} + \text{Width} \times \text{Depth} \] Calculating: \[ \text{External Surface Area} = (52 , \text{cm} \times 45 , \text{cm}) + (52 , \text{cm} \times 21 , \text{cm}) + (45 , \text{cm} \times 21 , \text{cm}) \]
Calculating each term:
- \(52 \times 45 = 2340 , \text{cm}^2\)
- \(52 \times 21 = 1092 , \text{cm}^2\)
- \(45 \times 21 = 945 , \text{cm}^2\)
Adding these together: \[ \text{External Surface Area} = 2340 + 1092 + 945 = 4377 , \text{cm}^2 \]
Internal Area Calculation: Now we calculate the internal surface area using the internal dimensions: \[ \text{Internal Surface Area} = \text{Internal Length} \times \text{Internal Width} + \text{Internal Length} \times \text{Internal Depth} + \text{Internal Width} \times \text{Internal Depth} \] Calculating: \[ \text{Internal Surface Area} = (51 , \text{cm} \times 44 , \text{cm}) + (51 , \text{cm} \times 20.5 , \text{cm}) + (44 , \text{cm} \times 20.5 , \text{cm}) \]
Calculating each term:
- \(51 \times 44 = 2244 , \text{cm}^2\)
- \(51 \times 20.5 = 1045.5 , \text{cm}^2\)
- \(44 \times 20.5 = 902 , \text{cm}^2\)
Adding these together: \[ \text{Internal Surface Area} = 2244 + 1045.5 + 902 = 4191.5 , \text{cm}^2 \]
Step 3: Calculate the Area of the Wood
Finally, the area of the wood used in the box is the difference between the external and internal surface areas: \[ \text{Area of the Wood} = \text{External Surface Area} - \text{Internal Surface Area} \] \[ \text{Area of the Wood} = 4377 , \text{cm}^2 - 4191.5 , \text{cm}^2 = 185.5 , \text{cm}^2 \]
Final Answer:
The area of the wood used to construct the open box is 185.5 cm².