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 using the information provided about the triangular face.
The triangular face has the following dimensions:
- Base \( b = 8 \) cm
- Height \( h = 6 \) cm
-
Calculate the Area of the Triangular Base: \[ \text{Area of the triangle} = \frac{1}{2} \times b \times h = \frac{1}{2} \times 8 , \text{cm} \times 6 , \text{cm} = 24 , \text{cm}^2 \]
-
Calculate the Perimeter of the Triangle: Since the triangle is isosceles with left and right sides each measuring \( 10 \) cm and the base measuring \( 8 \) cm, \[ \text{Perimeter} = 10 , \text{cm} + 10 , \text{cm} + 8 , \text{cm} = 28 , \text{cm} \]
-
Calculate the Surface Area of the Prism: The surface area \( S \) of the triangular prism can be found using the formula: \[ S = \text{Area of the two triangular bases} + \text{Area of the three rectangular faces} \]
The area of the two triangular bases is: \[ 2 \times \text{Area of triangle} = 2 \times 24 , \text{cm}^2 = 48 , \text{cm}^2 \]
The area of the rectangular sides (where \( l \) is the length of the prism):
- Area of the first rectangular face (base of triangle): \( 8 \times l \)
- Area of the second rectangular face (left side): \( 10 \times l \)
- Area of the third rectangular face (right side): \( 10 \times l \)
Therefore, the total area of the three rectangular faces is: \[ (8 + 10 + 10) \times l = 28l \]
Putting it all together: \[ S = 48 + 28l \]
We know the total surface area \( S \) is \( 524 \) cm²: \[ 48 + 28l = 524 \] \[ 28l = 524 - 48 \] \[ 28l = 476 \] \[ l = \frac{476}{28} \approx 17 \]
Thus, the missing length \( l \) of the triangular prism is \( \boxed{17 , \text{cm}} \).