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The problem involves finding the value of \( y \) for a triangular prism with a given surface area.

### Given Information:
1. The surface area of the triangular prism is \( 148 \, \text{cm}^2 \).
2. The dimensions of the triangular prism are provided:
   - The base of the triangular face is \( 5 \, \text{cm} \).
   - The height of the triangular face is \( 4 \, \text{cm} \).
   - The lengths of the three sides of the triangular face are \( 3 \, \text{cm} \), \( 5 \, \text{cm} \), and \( 4 \, \text{cm} \).
   - The length (height) of the prism is \( y \, \text{cm} \).

### Step-by-Step Solution:

To find \( y \), we'll use the formula for the surface area of a triangular prism:

\[
\text{Surface Area} = \text{Base Area} + \text{Lateral Area}
\]

1. **Calculate the Base Area:**
   The base of the triangular face is a right triangle with:
   - Base = \( 5 \, \text{cm} \)
   - Height = \( 4 \, \text{cm} \)

\[
\text{Base Area} = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 5 \times 4 = 10 \, \text{cm}^2
\]

Since the prism has two triangular bases, the total base area is:

\[
\text{Total Base Area} = 2 \times 10 = 20 \, \text{cm}^2
\]

2. **Calculate the Lateral Area:**
   The lateral area is the sum of the areas of the three rectangular faces formed by the sides of the triangle and the length \( y \) of the prism.

   - Side lengths are \( 3 \, \text{cm} \), \( 4 \, \text{cm} \), and \( 5 \, \text{cm} \).

   So, the lateral area is:

\[
\text{Lateral Area} = (3 \times y) + (4 \times y) + (5 \times y) = (3 + 4 + 5) y = 12y \, \text{cm}^2
\]

3. **Set up the Surface Area Equation:**

Given that the total surface area is \( 148 \, \text{cm}^2 \):

\[
\text{Total Surface Area} = \text{Base Area} + \text{Lateral Area}
\]

\[
148 = 20 + 12y
\]

4. **Solve for \( y \):**

\[
148 - 20 = 12y
\]

\[
128 = 12y
\]

\[
y = \frac{128}{12} = 10.\overline{6}
\]

### Conclusion:
The value of \( y \) is approximately \( 10.67 \). However, the closest whole number answer choice is:

\[
\boxed{C: 10}
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you find the surface area of a prism with different triangular bases?
2. What is the formula for the surface area of a cylinder?
3. How can you calculate the volume of a triangular prism?
4. What happens to the surface area when the height of the prism doubles?
5. How does the type of base affect the total surface area of a prism?

**Tip:** Always check if the base of the prism is a right triangle; this simplifies surface area calculations significantly.

In the diagram below, the surface area of the prism is 148 cm². Find the value of y.

Learn how to find the height of a triangular prism when the surface area is given as 148 cm². This problem involves using the formula for the surface area of a prism and calculating the base and lateral areas. Suitable for students in Grades 6-8.

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