To find the total surface area of the composite 3D figure made up of the three rectangular prisms (pedestals) described, we need to calculate the surface area of each individual prism and then account for the overlapping parts.
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Dimensions:
- For pedestal 1 (height = 15 inches): Length = 20 in, Depth = 20 in
- For pedestal 2 (height = 11 inches): Length = 20 in, Depth = 20 in
- For pedestal 3 (height = 7 inches): Length = 20 in, Depth = 20 in
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Surface Area of Each Prism: The surface area \( S \) of a rectangular prism can be calculated using the formula: \[ S = 2lw + 2lh + 2wh \] where \( l \) is the length, \( w \) is the depth (width), and \( h \) is the height.
Surface Area of Pedestal 1: \[ S_1 = 2(20)(20) + 2(20)(15) + 2(20)(20) \] \[ S_1 = 2(400) + 2(300) + 2(400) = 800 + 600 + 800 = 2200 \text{ in}^2 \]
Surface Area of Pedestal 2: \[ S_2 = 2(20)(20) + 2(20)(11) + 2(20)(20) \] \[ S_2 = 2(400) + 2(220) + 2(400) = 800 + 440 + 800 = 2040 \text{ in}^2 \]
Surface Area of Pedestal 3: \[ S_3 = 2(20)(20) + 2(20)(7) + 2(20)(20) \] \[ S_3 = 2(400) + 2(140) + 2(400) = 800 + 280 + 800 = 1880 \text{ in}^2 \]
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Total Surface Area Calculation: Now, we add the surface areas of all three pedestals, but we must subtract the areas of the overlap between them.
Each pedestal shares the top face area with the pedestal above it:
- Pedestal 1 (top face of 20 in x 20 in): 400 in² (no overlap, it's the top)
- Pedestal 2 shares with Pedestal 1: 400 in²
- Pedestal 3 shares with Pedestal 2: 400 in²
So, the total area of overlaps to subtract: Since each overlap area appears twice in the sum of \( S_1 + S_2 + S_3 \), we subtract it only once for each of the overlapping areas:
- Between pedestal 1 and pedestal 2 = 400 in²
- Between pedestal 2 and pedestal 3 = 400 in²
Total Overlap Area: 800 in²
Total Surface Area: \[ \text{Total Surface Area} = S_1 + S_2 + S_3 - \text{Total Overlap Area} \] \[ = 2200 + 2040 + 1880 - 800 = 4320 \text{ in}^2 \]
Thus, the total surface area of the stand is 4320 in².