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.