To find the surface area of the right trapezoidal prism, we need to calculate the surface area of the two trapezoidal bases and the four rectangular faces.
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Trapezoidal Base Area Calculation: The area \( A_t \) of a trapezoid can be calculated using the formula: \[ A_t = \frac{(b_1 + b_2) \cdot h}{2} \] where \( b_1 \) and \( b_2 \) are the lengths of the two bases and \( h \) is the height of the trapezoid.
In this case, the lengths of the bases are \( 2 \) cm and \( 6 \) cm, and the height of the trapezoid (which can be calculated from the side lengths) is given as \( 3 \) cm.
Let’s calculate the height of the trapezoid using the lengths provided. Typically, the height can be calculated from the dimensions, but since we're directly given the side lengths and height, let's proceed to calculate the area.
\[ \text{Area of base} = \frac{(2 \text{ cm} + 6 \text{ cm}) \cdot 3 \text{ cm}}{2} = \frac{8 \text{ cm} \cdot 3 \text{ cm}}{2} = 12 \text{ cm}^2 \]
Since there are two trapezoidal bases, we multiply this area by 2: \[ \text{Area of trapezoidal bases} = 2 \cdot 12 \text{ cm}^2 = 24 \text{ cm}^2 \]
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Rectangular Faces Calculation: Now, we calculate the area of the rectangular faces of the prism. The prism has:
- Two rectangular faces corresponding to the non-parallel sides of the trapezoid (lengths \( 3 \) cm and \( 5 \) cm).
- The sides have a height equal to the height of the prism \( 8 \) cm.
Calculate the area of each rectangle:
- For the side of length \( 3 \) cm: \[ \text{Area} = 3 \text{ cm} \cdot 8 \text{ cm} = 24 \text{ cm}^2 \]
- For the side of length \( 5 \) cm: \[ \text{Area} = 5 \text{ cm} \cdot 8 \text{ cm} = 40 \text{ cm}^2 \]
Plus, there are two faces of height \( 8 \) cm for the lengths of the bases \( 6 \) cm and \( 2 \) cm:
- For the base of length \( 6 \) cm: \[ \text{Area} = 6 \text{ cm} \cdot 8 \text{ cm} = 48 \text{ cm}^2 \]
- For the base of length \( 2 \) cm: \[ \text{Area} = 2 \text{ cm} \cdot 8 \text{ cm} = 16 \text{ cm}^2 \]
Therefore, the total area of all four rectangular faces is: \[ \text{Total area of rectangular faces} = 24 + 40 + 48 + 16 = 128 \text{ cm}^2 \]
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Total Surface Area: To find the total surface area \( A_s \) of the prism, we add the areas of the bases and the rectangular faces: \[ A_s = \text{Area of trapezoidal bases} + \text{Total area of rectangular faces} = 24 \text{ cm}^2 + 128 \text{ cm}^2 = 152 \text{ cm}^2 \]
Thus, the surface area of the right trapezoidal prism is \( 152 \text{ cm}^2 \).