To determine the total surface area of the triangular prism, we can start by calculating the areas of its various faces, including the triangular bases and the rectangular sides.
Step 1: Area of One Triangular Face
The triangular face is described as having an altitude of \(5 , \text{cm}\) and a base that likely corresponds to a side of the quadrilateral (since the sides of the quadrilateral are \(3 , \text{cm}\), \(4 , \text{cm}\), and \(6 , \text{cm}\)). The area of the triangle can be calculated using the formula:
\[ \text{Area of Triangle} = \frac{1}{2} \times \text{base} \times \text{height} \]
Assuming we take the base as \(6 , \text{cm}\):
\[ \text{Area of Triangle} = \frac{1}{2} \times 6 , \text{cm} \times 5 , \text{cm} = 15 , \text{cm}^2 \]
Step 2: Area of the Rectangular Faces
Assuming the prism has three rectangular faces corresponding to the three sides of the quadrilateral base (3 cm, 4 cm, 6 cm). The height of the prism isn't specified, but it does depend on inputs not provided directly in this problem.
If we denote the length of the prism (distance between the triangular bases) as \(h\), then the areas of the rectangular faces can be calculated as follows:
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Smallest Rectangular Face: \[ = \text{base} \times \text{height} = 3 , \text{cm} \times h = 3h , \text{cm}^2 \]
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Middle Rectangular Face: \[ = 4 , \text{cm} \times h = 4h , \text{cm}^2 \]
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Largest Rectangular Face: \[ = 6 , \text{cm} \times h = 6h , \text{cm}^2 \]
Step 3: Total Surface Area Calculation
The total surface area of the triangular prism can now be calculated using:
\[ \text{Total Surface Area} = 2 \times \text{Area of Triangle} + \text{Area of Rectangles} \]
Assuming we have the three rectangular faces:
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Area of two triangular faces: \[ = 2 \times 15 = 30 , \text{cm}^2 \]
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Area of all rectangular faces: \[ = 3h + 4h + 6h = 13h , \text{cm}^2 \]
So the Total Surface Area becomes:
\[ \text{Total Surface Area} = 30 + 13h , \text{cm}^2 \]
Conclusion
To sum it up:
- Area of ONE triangular face: \(15 , \text{cm}^2\)
- Area of the smallest rectangular face: \(3h , \text{cm}^2\)
- Area of the middle rectangular face: \(4h , \text{cm}^2\)
- Area of the largest rectangular face: \(6h , \text{cm}^2\)
- Total surface area of the triangular prism: \(30 + 13h , \text{cm}^2\)
Without the specific height \(h\), we can't provide a numerical answer for the total surface area. If a value for \(h\) is provided, you can substitute it in the final equation to find the total surface area.