To find the missing length \( L \) of the triangular prism, we'll first need to calculate the surface area of the prism provided that the surface area is 524 square centimeters.
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Calculate the area of the triangular base: The formula for the area \( A \) of a triangle is: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] Given that the base of the triangle is 8 cm and the height is 6 cm: \[ A = \frac{1}{2} \times 8 \times 6 = 24 \text{ cm}^2 \]
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Calculate the lateral surface area: A triangular prism has three rectangular lateral faces. The rectangles' dimensions are as follows:
- The height of the triangle corresponds to the two rectangular faces with dimensions (10 cm, L) (for each side of the triangle).
- The base of the triangle corresponds to the rectangular face with dimensions (8 cm, L).
So, the areas of the rectangular faces are:
- Two identical rectangles with dimensions \( 10 \text{ cm} \times L \): \[ 2 \times (10 \times L) = 20L \text{ cm}^2 \]
- One rectangle with dimensions \( 8 \text{ cm} \times L \): \[ 8L \text{ cm}^2 \]
Thus, the total lateral surface area is: \[ 20L + 8L = 28L \text{ cm}^2 \]
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Calculate the total surface area of the prism: The total surface area \( S \) of the prism is the sum of the lateral surface area and the area of the two triangular bases (twice the area of one triangle): \[ S = 28L + 2 \times \text{area of triangle} \] Since we calculated the area of one triangle to be 24 cm²: \[ S = 28L + 2 \times 24 \] \[ S = 28L + 48 \]
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Set up the equation for the total surface area: Given that the surface area is 524 cm²: \[ 28L + 48 = 524 \]
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Solve for \( L \): First, subtract 48 from both sides: \[ 28L = 524 - 48 \] \[ 28L = 476 \] Now, divide by 28: \[ L = \frac{476}{28} \approx 17 \]
Thus, the missing length \( L \) of the triangular prism is \( \boxed{17 \text{ cm}} \).