To find the total surface area of the composite figure formed by the two rectangular prisms, we will calculate the surface area of each prism separately and then sum them, making sure to account for the shared face between the two prisms.
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Dimensions of the prisms:
- Left rectangular prism:
- Length \( = 10 \) mm
- Height \( = 8 \) mm
- Width \( = w \) mm (we will determine this based on the alignment with the other prism)
- Right rectangular prism:
- Length \( = 6 \) mm
- Width \( = 6 \) mm
- Height \( = 6 \) mm
- Left rectangular prism:
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Combining the width: Since the two prisms align perfectly and considering the combined length of both prisms is 16 mm, the width of the left prism must also be \( = 6 \) mm (to align with the right prism).
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Surface area calculations:
- The formula for the surface area of a rectangular prism is given by: \[ SA = 2(lw + lh + wh) \]
- Surface Area of Left Rectangular Prism: \[ SA_{\text{left}} = 2(10 \cdot 6 + 10 \cdot 8 + 6 \cdot 8) = 2(60 + 80 + 48) = 2 \cdot 188 = 376 \text{ mm}^2 \]
- Surface Area of Right Rectangular Prism: \[ SA_{\text{right}} = 2(6 \cdot 6 + 6 \cdot 6 + 6 \cdot 6) = 2(36 + 36 + 36) = 2 \cdot 108 = 216 \text{ mm}^2 \]
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Accounting for the shared face: The two prisms share one face, which has dimensions equal to the width of the right prism and the height of the left prism:
- Shared face area \( = 6 \cdot 6 = 36 \) mm².
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Total Surface Area of the composite figure: \[ SA_{\text{total}} = SA_{\text{left}} + SA_{\text{right}} - \text{Shared Area} \] \[ SA_{\text{total}} = 376 + 216 - 36 = 556 \text{ mm}^2 \]
The total surface area of the figure is 556 mm².