To find the missing length \( l \) of the triangular prism given that the surface area is 524 square centimeters, we first need to calculate the surface area of the prism.
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Calculate the area of the triangular base:
- The base of the triangle measures 8 cm, and the height from the top vertex to the base measures 6 cm.
- Area of the triangle \( A = \frac{1}{2} \times \text{base} \times \text{height} \): \[ A = \frac{1}{2} \times 8 , \text{cm} \times 6 , \text{cm} = \frac{48}{2} , \text{cm}^2 = 24 , \text{cm}^2 \]
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Calculate the surface area of the prism:
The surface area \( SA \) of a triangular prism is given by: \[ SA = 2 \times \text{Area of the triangular base} + \text{Perimeter of the triangular base} \times \text{Height of the prism} \]
- The perimeter \( P \) of the triangle is calculated as: \[ P = 8 , \text{cm} + 10 , \text{cm} + 10 , \text{cm} = 28 , \text{cm} \]
- Substituting into the surface area formula: \[ SA = 2 \times 24 , \text{cm}^2 + 28 , \text{cm} \times l \] \[ SA = 48 , \text{cm}^2 + 28l \]
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Set the equation equal to the given surface area: \[ 48 , \text{cm}^2 + 28l = 524 , \text{cm}^2 \] \[ 28l = 524 - 48 \] \[ 28l = 476 \] \[ l = \frac{476}{28} = 17 \]
Thus, the missing length \( l \) of the triangular prism is \( 17 , \text{cm} \).
The answer is: 17 cm