To calculate the total surface area of the composite figure formed by the two rectangular prisms, we first need to find the surface area of each prism and then combine them, while accounting for the overlapping area where they connect.
Step 1: Identify dimensions.
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Left rectangular prism (smaller prism):
- Length = 10 mm
- Height = 8 mm
- Width (W) = (since the combined length is 16 mm and the right prism has a length of 6 mm):
- Width = 16 mm - 10 mm = 6 mm
-
Right rectangular prism (larger prism):
- Length = 6 mm
- Width = 6 mm
- Height = 6 mm
Step 2: Calculate the surface area of each prism.
The surface area (SA) of a rectangular prism is given by: \[ SA = 2(lw + lh + wh) \]
Surface area of the left rectangular prism:
- Length \(l = 10 \text{ mm}\)
- Width \(w = 6 \text{ mm}\)
- Height \(h = 8 \text{ mm}\)
Calculating: \[ SA_{left} = 2(10 \times 6 + 10 \times 8 + 6 \times 8) \] \[ = 2(60 + 80 + 48) \] \[ = 2(188) = 376 \text{ square mm} \]
Surface area of the right rectangular prism:
- Length \(l = 6 \text{ mm}\)
- Width \(w = 6 \text{ mm}\)
- Height \(h = 6 \text{ mm}\)
Calculating: \[ SA_{right} = 2(6 \times 6 + 6 \times 6 + 6 \times 6) \] \[ = 2(36 + 36 + 36) \] \[ = 2(108) = 216 \text{ square mm} \]
Step 3: Find the overlapping area.
The two prisms overlap in the area where they connect. This area is the top face of the right prism, which has dimensions 6 mm by 6 mm: \[ \text{Overlapping area} = 6 \times 6 = 36 \text{ square mm} \]
Step 4: Calculate total surface area.
Now, we combine the surface areas of both prisms and subtract the overlapping area: \[ \text{Total Surface Area} = SA_{left} + SA_{right} - \text{Overlapping area} \] \[ = 376 + 216 - 36 \] \[ = 556 \text{ square mm} \]
Thus, the total surface area of the figure is 556 square millimeters.