To find the surface area of the box that needs to be decorated, we first calculate the total surface area of the box and then subtract the area of the bottom and the back.
The dimensions of the box are:
- Length (L) = 27 cm
- Width (W) = 20 cm
- Height (H) = 20 cm
The surface area \( A \) of a rectangular box is calculated using the formula: \[ A = 2(LW + LH + WH) \]
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Calculate the area of each pair of sides:
- \( LW = 27 \times 20 = 540 \) cm² (front and back)
- \( LH = 27 \times 20 = 540 \) cm² (left and right sides)
- \( WH = 20 \times 20 = 400 \) cm² (top and bottom)
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Calculate total surface area: \[ A = 2(540 + 540 + 400) = 2(1480) = 2960 \text{ cm}² \]
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Now, we subtract the areas of the bottom and the back:
- Area of the bottom (W x L) = \( 20 \times 27 = 540 \) cm²
- Area of the back (H x W) = \( 20 \times 20 = 400 \) cm²
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Calculate the total area to be subtracted: \[ \text{Total area to subtract} = 540 + 400 = 940 \text{ cm}² \]
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Finally, the surface area that needs to be decorated: \[ \text{Decorated surface area} = 2960 - 940 = 2020 \text{ cm}² \]
The surface area of the box that needs to be decorated is 2020 cm².