To find the missing length \(l\) of the triangular prism given that its surface area is 524 square centimeters, we need to calculate the surface area of the prism using the provided dimensions.
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Calculate the area of the triangular face:
- The base of the triangle \(b = 8\) cm.
- The height of the triangle \(h = 6\) cm.
- The area \(A\) of the triangle is given by the formula: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
- Substituting the values we have: \[ A = \frac{1}{2} \times 8 \times 6 = 24 \text{ cm}^2 \]
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Calculate the total area of the two triangular faces:
- Since there are two triangular faces, we multiply the area of one triangle by 2: \[ \text{Total area of triangular faces} = 2 \times 24 = 48 \text{ cm}^2 \]
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Calculate the area of the rectangular faces:
- There are three rectangular faces in the prism:
- Two rectangles corresponding to the sides of the triangle (10 cm each) and the length \(l\).
- Area of one rectangle \( = \text{height} \times \text{length} = 10 \times l\)
- Since there are two of these rectangles, their combined area is: \[ 2 \times (10 \times l) = 20l \]
- One rectangle corresponding to the base of the triangle:
- Area of this rectangle \( = \text{base} \times \text{length} = 8 \times l\)
- Two rectangles corresponding to the sides of the triangle (10 cm each) and the length \(l\).
- There are three rectangular faces in the prism:
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Combine all areas for total surface area:
- The total surface area \(S\) can be expressed as: \[ S = \text{Area of triangular faces} + \text{Area of rectangular faces} \]
- Substituting the values we have: \[ S = 48 + 20l + 8l = 48 + 28l \]
- According to the problem, the surface area is given as 524 cm²: \[ 48 + 28l = 524 \]
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Solve for \(l\):
- Subtract 48 from both sides: \[ 28l = 524 - 48 \] \[ 28l = 476 \]
- Divide both sides by 28: \[ l = \frac{476}{28} \] \[ l = 17 \]
Thus, the missing length \(l\) of the triangular prism is 17 cm.