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The provided image shows a right-angled triangular prism. We are tasked with calculating:

1. The **volume** of the prism.
2. The **surface area** of the prism.

### Given Data:
- **Right triangle** ABC forms the base of the prism.
- \( AB = 3 \, \text{cm} \) (one leg of the right triangle).
- \( \angle ABC = 42^\circ \).
- The prism has a length (distance between the two triangular faces) of \( 10 \, \text{cm} \).

### Steps to Solve:

#### 1. **Volume of the Prism**:

The volume of a prism is given by the formula:
\[
\text{Volume} = \text{Base Area} \times \text{Length}
\]

For this triangular prism, we first need to calculate the area of the triangular base \( \triangle ABC \).

- To find the height \( BC \) (opposite the \( 42^\circ \) angle), we use the trigonometric relation:
\[
\sin(42^\circ) = \frac{BC}{AB}
\]
Thus, 
\[
BC = AB \times \sin(42^\circ) = 3 \times \sin(42^\circ)
\]
- Similarly, to find the adjacent side \( AC \), use:
\[
AC = AB \times \cos(42^\circ) = 3 \times \cos(42^\circ)
\]
  
Once we have both \( BC \) and \( AC \), the area of triangle \( ABC \) is:
\[
\text{Area of } \triangle ABC = \frac{1}{2} \times AB \times BC
\]

Finally, multiplying this base area by the prism's length (10 cm) gives the volume.

#### 2. **Surface Area of the Prism**:

The surface area of the prism consists of:
- Two triangular faces \( ABC \) and \( EDC \).
- Three rectangular faces: \( ABED \), \( BCDE \), and \( ACDE \).

We calculate the areas of these individual faces:
- The area of the triangular faces \( ABC \) is already calculated.
- The areas of the three rectangular faces are calculated using the sides of the triangle and the prism's length:
  - \( ABED \) has dimensions \( AB \times 10 \).
  - \( BCDE \) has dimensions \( BC \times 10 \).
  - \( ACDE \) has dimensions \( AC \times 10 \).

Adding all the areas gives the total surface area of the prism.

Let me calculate these values for you.### Solutions:

1. **Volume of the prism**: \( 30.11 \, \text{cm}^3 \)
2. **Surface area of the prism**: \( 78.39 \, \text{cm}^2 \)

Would you like further clarification on any steps or additional details?

Here are 5 related questions for further practice:
1. How would the volume change if the length of the prism were doubled?
2. Can you calculate the height of the triangle \( BC \) in terms of the prism's dimensions?
3. How would the surface area change if the angle \( \angle ABC \) were increased?
4. What would the area of one of the rectangular faces be if the length of the prism were increased by 5 cm?
5. How would you calculate the volume of the prism if the base were not a right triangle?

**Tip**: When calculating the surface area of a prism, always ensure to account for all faces, including the base and lateral faces.

Calculate the volume and surface area of the given triangular prism with a right triangle base ABC, where AB = 3 cm, angle ABC = 42°, and the length of the prism is 10 cm.

Learn how to calculate the volume and surface area of a triangular prism with a right triangle base. Given: AB = 3 cm, angle ABC = 42°, and length of the prism is 10 cm. Using trigonometric functions, find the dimensions and compute the total surface area and volume.

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