To find the surface area of the net of a triangular prism, we need to calculate the area of all its faces, which include two triangular bases and three rectangular faces.
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Calculating the Area of the Triangular Faces: The triangular face has sides of lengths 17 m, 17 m, and 16 m. We can use Heron's formula to find the area of the triangle.
First, we find the semi-perimeter \( s \): \[ s = \frac{17 + 17 + 16}{2} = 25 \]
Now, we apply Heron's formula: \[ A = \sqrt{s(s-a)(s-b)(s-c)} \] where \( a = 17 \), \( b = 17 \), and \( c = 16 \): \[ A = \sqrt{25 \times (25 - 17) \times (25 - 17) \times (25 - 16)} \] \[ = \sqrt{25 \times 8 \times 8 \times 9} \] \[ = \sqrt{25 \times 64 \times 9} \] \[ = \sqrt{14400} = 120 , \text{m}^2 \] Since there are 2 triangular faces: \[ \text{Area of both triangular faces} = 2 \times 120 = 240 , \text{m}^2 \]
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Calculating the Area of the Rectangular Faces: We know the dimensions of the rectangular faces:
- The rectangles are formed between each side of the triangle and the length of the prism.
- The lengths of the prism will be the height (or side) of the triangle, which is given as 15 m.
We have three rectangular faces with the following dimensions:
- One rectangle (base) with dimensions \( 16 \times 15 \) m
- Two rectangles (sides) with dimensions \( 17 \times 15 \) m for each side.
Now, calculating the area of each rectangular face:
- Area of the base rectangle: \[ A_1 = 16 \times 15 = 240 , \text{m}^2 \]
- Area of one of the side rectangles: \[ A_2 = 17 \times 15 = 255 , \text{m}^2 \]
- Area of the other side rectangle (same dimensions): \[ A_3 = 17 \times 15 = 255 , \text{m}^2 \]
Now, sum up the areas of the rectangular faces: \[ \text{Total area of rectangular faces} = A_1 + A_2 + A_3 = 240 + 255 + 255 = 750 , \text{m}^2 \]
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Total Surface Area of the Prism: Finally, we sum the areas of the triangular faces and rectangular faces: \[ \text{Total Surface Area} = \text{Area of triangular faces} + \text{Area of rectangular faces} \] \[ = 240 + 750 = 990 , \text{m}^2 \]
Thus, the total surface area of the net of the triangular prism is \( \boxed{990} \) square meters.