To find the missing length \( l \) of the triangular prism, we first need to calculate the area of the triangular base and then use this to determine the surface area of the prism.
Step 1: 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. Plugging in these values gives: \[ A = \frac{1}{2} \times 8 \times 6 = \frac{1}{2} \times 48 = 24 \text{ cm}^2 \]
Step 2: Calculate the surface area of the prism.
The surface area \( S \) of a triangular prism is given by the formula: \[ S = 2A + \text{perimeter} \times l \] where \( l \) is the length of the prism and \( A \) is the area of the triangular base.
Step 3: Calculate the perimeter of the triangular base.
The sides of the triangle are \( 10 \) cm, \( 10 \) cm, and \( 8 \) cm. So the perimeter \( P \) is: \[ P = 10 + 10 + 8 = 28 \text{ cm} \]
Step 4: Substitute values into the surface area formula.
We know that the surface area \( S \) is \( 524 \) cm² and the area \( A \) is \( 24 \) cm²: \[ 524 = 2 \times 24 + 28 \times l \] This simplifies to: \[ 524 = 48 + 28l \] Subtracting \( 48 \) from both sides: \[ 476 = 28l \]
Step 5: Solve for \( l \).
Now, divide both sides by \( 28 \): \[ l = \frac{476}{28} = 17 \text{ cm} \]
Thus, the missing length \( l \) of the triangular prism is \( 17 \) cm.
The correct answer is: 17 cm.