To find the total surface area of the composite figure formed by the two rectangular prisms, we need to calculate the surface area of each prism separately and then combine them while taking into account their overlapping areas.
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Smaller Rectangular Prism:
- Length (L) = 10 mm
- Height (H) = 8 mm
- Width (W) = to be determined.
Since the total length of the two prisms is 16 mm and the right prism has a length of 6 mm, we can deduce that:
- Width of the smaller rectangular prism = Total length - Width of right prism = 16 mm - 6 mm = 10 mm.
The surface area (SA) of a rectangular prism is given by the formula: \[ SA = 2(LW + LH + WH) \] Plugging the dimensions: \[ SA = 2(10 \cdot 10 + 10 \cdot 8 + 10 \cdot 8) = 2(100 + 80 + 80) = 2(260) = 520 \text{ mm}^2 \]
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Larger Rectangular Prism:
- Length = 6 mm
- Width = 6 mm
- Height = 6 mm
Using the same surface area formula: \[ SA = 2(LW + LH + WH) = 2(6 \cdot 6 + 6 \cdot 6 + 6 \cdot 6) = 2(36 + 36 + 36) = 2(108) = 216 \text{ mm}^2 \]
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Accounting for Overlap: When adding the areas, we need to subtract the overlapping area where the two prisms connect:
- The overlapping area is equal to the area of the face of the smaller rectangular prism that connects with the larger one. This face has a length of 6 mm (the width of the larger prism) and a height of 8 mm (the height of the smaller prism).
- Overlapping area = Width * Height = 6 mm * 8 mm = 48 mm².
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Total Surface Area: \[ \text{Total Surface Area} = SA_{smaller} + SA_{larger} - \text{overlapping area} \] \[ = 520 \text{ mm}^2 + 216 \text{ mm}^2 - 48 \text{ mm}^2 \] \[ = 688 \text{ mm}^2 \]
Therefore, the total surface area of the composite figure is 688 mm².