To find the total surface area of the composite figure formed by the two rectangular prisms, we will first determine the surface area of each prism separately and then subtract the area of the connecting face that is not exposed.
Step 1: Surface Area of the Smaller Rectangular Prism
The dimensions of the smaller rectangular prism are:
- Length = 10 mm
- Height = 8 mm
- Width = W mm (not given specifically, but will be determined)
The surface area (SA) of a rectangular prism can be calculated using the formula:
\[ SA = 2(lw + lh + wh) \]
where \(l\) is length, \(w\) is width, and \(h\) is height. For the smaller prism, substituting the known dimensions:
\[ SA_{small} = 2(10W + 10 \cdot 8 + W \cdot 8) = 2(10W + 80 + 8W) = 2(18W + 80) \]
\[ SA_{small} = 36W + 160 \quad \text{(1)} \]
Step 2: Surface Area of the Larger Rectangular Prism
The dimensions of the larger rectangular prism are:
- Length = 6 mm
- Height = 6 mm
- Width = 6 mm
Calculating the surface area:
\[ SA_{large} = 2(6 \cdot 6 + 6 \cdot 6 + 6 \cdot 6) = 2(36 + 36 + 36) = 2(108) = 216 \quad \text{(2)} \]
Step 3: Subtracting the Overlapping Face Area
The two prisms overlap at their width. Since the width of both prisms must be equal (because they align perfectly), let's denote W as the width of the smaller prism.
The area of the overlapping face is determined by:
- Height = 6 mm (for the larger prism)
- Width = W mm
So, the area of the overlapping face:
\[ Area_{overlap} = W \cdot 6 \]
Step 4: Total Surface Area Calculation
Combining the surface areas and subtracting the overlap:
\[ Total , SA = SA_{small} + SA_{large} - Area_{overlap} \]
Substituting from (1) and (2):
\[ Total , SA = (36W + 160) + 216 - 6W = 30W + 376 \]
Step 5: Width W Calculation
From the problem, the total length of both combined prisms is given as 16 mm, and since the left prism is 10 mm and the right prism is 6 mm, and they meet perfectly at their widths, we don’t need to know W to compute total dimensions fully, but functionally if \(W\) must be known for precise calculation. However, let's assume W according to overlap perception stays unnumbered in this form.
Total Surface Area Result
The expression “30W + 376” signifies total surface area contingent upon specific determination or observation measurement of W, typically concluding without specific W, the last answer remains in its given functional mathematical form.
Thus, if dimensional width specifics are gathered (like W=6 or calculated), simply substitute for final answer total area accordingly.
If \(W\) was known to be 6 mm:
Find \( Total , SA\) \[ SA_{Total} = 30 \cdot 6 + 376 = 180 + 376 = 556 , mm^2 \] The total surface area of the figure is thus \(556 , mm^2\) if W=6 mm. Else factual depends on specific W (Width) as determined or provided for assumptions.
Final answer in conditional function to be structured into specific presented output dispositionally with numeric illustration transparency for width reference guidance in visual dimensions trimmed on necessary ask.