To find the surface area of the steamer basket, we need to consider both the areas of the circular bases and the area of the rectangular face connecting the circles.
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Area of the two circular bases: Since there are two bases, we multiply the area of one base by 2: \[ \text{Area of circular bases} = 2 \times 452.16 , \text{cm}^2 = 904.32 , \text{cm}^2 \]
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Area of the rectangular face: The area of the rectangular face is given as: \[ \text{Area of rectangular face} = 678.24 , \text{cm}^2 \]
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Total surface area: Now we add these two areas together to find the total surface area of the steamer basket: \[ \text{Total Surface Area} (SA) = \text{Area of circular bases} + \text{Area of rectangular face} \] \[ SA = 904.32 , \text{cm}^2 + 678.24 , \text{cm}^2 = 1582.56 , \text{cm}^2 \]
Therefore, the surface area of the steamer basket is \( \boxed{1582.56 , \text{cm}^2} \).