To find the missing length \( l \) of the triangular prism when the surface area is given, we can follow these steps:
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Calculate the area of the triangular base: The area \( A \) of a triangle can be calculated using the formula: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] In this case, the base of the triangle is 8 cm and the height is 6 cm: \[ A = \frac{1}{2} \times 8 \times 6 = 24 \text{ square centimeters} \]
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Calculate the lateral surface area of the prism: The lateral surface area of a triangular prism consists of three rectangular faces. The area of each rectangular face is calculated as follows:
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The area of the rectangle corresponding to the base (which is 8 cm) is: \[ A_{base} = \text{base} \times l = 8 \times l \]
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The area of the rectangle corresponding to the left side of the triangle (which is 10 cm) is: \[ A_{side1} = \text{side1} \times l = 10 \times l \]
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The area of the rectangle corresponding to the right side of the triangle (which is also 10 cm) is: \[ A_{side2} = \text{side2} \times l = 10 \times l \]
Adding these areas together, we get: \[ A_{\text{lateral}} = A_{base} + A_{side1} + A_{side2} = (8 \times l) + (10 \times l) + (10 \times l) = (8 + 10 + 10)l = 28l \]
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Calculate the total surface area: The total surface area \( S \) of the prism is the sum of the area of the two triangular bases and the lateral surface area: \[ S = 2 \times A + A_{\text{lateral}} = 2 \times 24 + 28l = 48 + 28l \]
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Set up the equation with the given surface area: Set the total surface area equal to 524: \[ 48 + 28l = 524 \]
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Solve for \( l \): Subtract 48 from both sides: \[ 28l = 524 - 48 \] \[ 28l = 476 \] Divide both sides by 28: \[ l = \frac{476}{28} = 17 \]
Thus, the missing length \( l \) of the triangular prism is 17 cm.