To find the total surface area of the composite figure, we need to calculate the surface area of each rectangular prism and then adjust for any faces that are hidden where they connect.
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Surface Area of the Smaller Rectangular Prism:
- Given dimensions: length = 10 mm, height = 8 mm, width = W (unknown).
- We can find the width by knowing that the combined length of the two prisms is 16 mm, so: \[ 10 + 6 = 16 \] This means the width of the smaller prism aligns with the width of the larger prism, which is 6 mm.
- Hence, the dimensions of the smaller prism are: length = 10 mm, width = 6 mm, height = 8 mm.
- The surface area formula for a rectangular prism is: \[ SA = 2lw + 2lh + 2wh \]
- Plugging in the dimensions for the smaller prism: \[ SA = 2(10 \times 6) + 2(10 \times 8) + 2(6 \times 8) = 2(60) + 2(80) + 2(48) = 120 + 160 + 96 = 376 \text{ mm}^2 \]
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Surface Area of the Larger Rectangular Prism:
- Given dimensions: length = 6 mm, width = 6 mm, height = 6 mm.
- Calculating the surface area: \[ SA = 2lw + 2lh + 2wh = 2(6 \times 6) + 2(6 \times 6) + 2(6 \times 6) = 2(36) + 2(36) + 2(36) = 72 + 72 + 72 = 216 \text{ mm}^2 \]
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Adjusting for Hidden Faces:
- The face where the smaller prism connects with the larger prism does not contribute to the external surface area. The area of that face is: \[ \text{Hidden Face Area} = \text{width} \times \text{height} = 6 \times 6 = 36 \text{ mm}^2 \]
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Calculating the Total Surface Area:
- The total surface area is then the sum of the surface areas of the two prisms minus the area of the hidden face: \[ \text{Total Surface Area} = \text{Surface Area of Smaller Prism} + \text{Surface Area of Larger Prism} - \text{Hidden Face Area} \] \[ = 376 + 216 - 36 = 556 \text{ mm}^2 \]
Thus, the total surface area of the figure is 556 mm².