To calculate the missing length "I" of the triangular prism given the total surface area, we need to analyze the provided dimensions and determine the surface area of the prism.
Given the prism's dimensions:
- Base of the triangular face (bottom) = 8 cm
- Height of the triangular face = 10 cm (assuming this is the vertical height from the base to the apex)
- Side of the triangular face = 6 cm
The area of the triangular face can be calculated using the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Thus, the area of the triangular face is:
\[ \text{Area}_{\triangle} = \frac{1}{2} \times 8 \times 10 = 40 , \text{cm}^2 \]
Since there are two triangular faces in the prism, we multiply this area by 2:
\[ \text{Area of both triangular faces} = 2 \times 40 = 80 , \text{cm}^2 \]
Next, we need to calculate the lateral surface area of the prism. The lateral surface area consists of three rectangular faces. We'll denote the height of the prism (which is not given directly) as "I".
- The first rectangular face has dimensions 8 cm (base of triangle) and I, hence its area is \(8 \times I\).
- The second rectangular face has dimensions 10 cm (side of triangle) and I, hence its area is \(10 \times I\).
- The third rectangular face has dimensions 6 cm (other side of triangle) and I, hence its area is \(6 \times I\).
Now, we can express the total lateral surface area as:
\[ \text{Area}_{lateral} = (8 + 10 + 6) \times I = 24I \]
Now we can sum the total surface area:
\[ \text{Total Surface Area} = \text{Area of both triangular faces} + \text{Area}_{lateral} \]
Substituting the values:
\[ 524 = 80 + 24I \]
Now, we solve for I:
\[ 524 - 80 = 24I \] \[ 444 = 24I \] \[ I = \frac{444}{24} \approx 18.5 \]
Since 18.5 cm isn't among the answer choices (A. 16 cm, B. 8 cm, C. 17 cm, D. 10 cm), and reviewing the calculations, it seems that either some dimensions are incorrect or the answer choices provided are not matching the calculations based on the given surface area.
However, since we have to choose the closest value in the provided options to match a typical real-world scenario, 16 cm (Option A) is a plausible selection if rounding or estimation is considered.
Thus, my conclusion is:
A. 16 cm