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Let's solve each problem step-by-step.

### Problem 7

We need to calculate the volume of the prism.

The prism's volume \( V \) can be found using the formula:
\[
V = \text{Area of cross-section} \times \text{Length (or height)}
\]

Given:
- Area of cross-section = 42 m²
- Length (or height) = 8 m

**Calculation:**
\[
V = 42 \, \text{m}^2 \times 8 \, \text{m} = 336 \, \text{m}^3
\]

**Answer for Problem 7:** The volume of the prism is **336 m³**.

### Problem 8

We need to find the volume of each prism by first calculating the area of the cross-section.

#### Problem 8a

**Shape**: Triangular Prism

**Dimensions given**:
- Base of triangle = 8 cm
- Height of triangle = 6 cm
- Length of prism = 10 cm

**Steps**:
1. **Calculate the area of the triangle cross-section** using the formula:
   \[
   \text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height}
   \]
   \[
   \text{Area} = \frac{1}{2} \times 8 \, \text{cm} \times 6 \, \text{cm} = 24 \, \text{cm}^2
   \]

2. **Calculate the volume of the prism** using the formula:
   \[
   V = \text{Area of cross-section} \times \text{Length}
   \]
   \[
   V = 24 \, \text{cm}^2 \times 10 \, \text{cm} = 240 \, \text{cm}^3
   \]

**Answer for Problem 8a:** The volume of the prism is **240 cm³**.

#### Problem 8b

**Shape**: Trapezoidal Prism

**Dimensions given**:
- Top base of trapezoid = 4 m
- Bottom base of trapezoid = 8 m
- Height of trapezoid = 5 m
- Length of prism = 12 m

**Steps**:
1. **Calculate the area of the trapezoid cross-section** using the formula:
   \[
   \text{Area} = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height}
   \]
   \[
   \text{Area} = \frac{1}{2} \times (4 \, \text{m} + 8 \, \text{m}) \times 5 \, \text{m}
   \]
   \[
   \text{Area} = \frac{1}{2} \times 12 \, \text{m} \times 5 \, \text{m} = 30 \, \text{m}^2
   \]

2. **Calculate the volume of the prism** using the formula:
   \[
   V = \text{Area of cross-section} \times \text{Length}
   \]
   \[
   V = 30 \, \text{m}^2 \times 12 \, \text{m} = 360 \, \text{m}^3
   \]

**Answer for Problem 8b:** The volume of the prism is **360 m³**.

---

Would you like more details on any of these steps, or do you have any questions? 

### Related Questions:
1. How do you find the area of a circle and use it to calculate the volume of a cylindrical prism?
2. What is the formula for finding the surface area of a prism?
3. How does the volume formula change for prisms with different cross-sectional shapes?
4. Can you explain how to find the volume of a pyramid and how it differs from finding the volume of a prism?
5. How would you calculate the volume of a prism with a polygonal cross-section like a pentagon or hexagon?

**Tip:** When calculating volumes, always ensure your cross-sectional area is correctly calculated based on the specific shape of the prism.

Learn how to calculate the volume of prisms with simple geometric shapes. This problem covers basic principles of geometry, focusing on the volume formula for prisms. Suitable for students in Grades 5-7.

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