To find the surface area of Nikolai's L-shaped platform, we need to break the shape down into a few simple parts.
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Calculate the full area of the larger rectangle:
- Length = 16 inches
- Width = 10 inches \[ \text{Area}_{\text{large}} = \text{length} \times \text{width} = 16 , \text{in.} \times 10 , \text{in.} = 160 , \text{in.}^2 \]
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Calculate the area of the removed rectangle:
- The length of the removed portion = 12 inches
- The width of the removed portion = 4 inches \[ \text{Area}_{\text{removed}} = \text{length} \times \text{width} = 12 , \text{in.} \times 4 , \text{in.} = 48 , \text{in.}^2 \]
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Calculate the area of the L-shaped platform:
- The area of the L-shaped platform is the area of the larger rectangle minus the area of the removed rectangle. \[ \text{Area}{\text{L-shape}} = \text{Area}{\text{large}} - \text{Area}_{\text{removed}} = 160 , \text{in.}^2 - 48 , \text{in.}^2 = 112 , \text{in.}^2 \]
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Calculate the surface area of the 3D platform: Since the platform is 3 inches thick, we will consider both the top and the bottom surfaces, as well as the sides.
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Top surface area: This is the area calculated above.
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Bottom surface area: This is the same as the top surface area, so also \(112 , \text{in.}^2\).
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Sides: The platform has a thickness of \(3\) inches, and we will have to calculate the area of the sides. The L-shaped platform has four sides:
- Two longer sides (length 16 inches): \[ \text{Area}_{\text{side1}} = \text{length} \times \text{thickness} \times 2 = 3 , \text{in.} \times 16 , \text{in.} \times 2 = 96 , \text{in.}^2 \]
- Two shorter sides (which are 10 inches wide on one side and 4 inches on the side with the removed portion): \[ \text{Area}_{\text{side2}} = \text{thickness} \times \text{width1} + \text{thickness} \times \text{width2} = (3 \times 10) + (3 \times 4) = 30 + 12 = 42 , \text{in.}^2 \]
- Add all the surface areas together: \[ \text{Total Surface Area} = \text{Area}{\text{top}} + \text{Area}{\text{bottom}} + \text{Area}_{\text{sides}} = 112 + 112 + (96 + 42) = 112 + 112 + 138 = 362 , \text{in.}^2 \]
Therefore, the surface area of the platform is \(362 , \text{in.}^2\).