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### Question 4.1: Rectangular Prism
We are given a rectangular prism with:
- Volume \( V = 270 \, \text{cm}^3 \)
- Length \( l = 15 \, \text{cm} \)
- Breadth \( b = 6 \, \text{cm} \)
- Height \( h \) is unknown.

The volume of a rectangular prism is given by:
\[
V = l \times b \times h
\]
Substitute the known values:
\[
270 = 15 \times 6 \times h
\]
Simplify the right-hand side:
\[
270 = 90 \times h
\]
Solve for \( h \):
\[
h = \frac{270}{90} = 3 \, \text{cm}
\]

Thus, the height \( h \) of the rectangular prism is **3 cm**.

---

### Question 4.2: Cylinder Calculations
The cylinder has:
- Radius \( r = 3 \, \text{cm} \)
- Height \( h = 8 \, \text{cm} \)
- Use \( \pi = 3.142 \)

#### 4.2.1: Identify the shape
The shape is a **cylinder**.

#### 4.2.2: How many faces does it have?
A cylinder has **3 faces**:
- 2 circular faces (top and bottom)
- 1 curved lateral surface

#### 4.2.3: Volume of the cylinder
The volume of a cylinder is given by:
\[
V = \pi r^2 h
\]
Substitute the values:
\[
V = 3.142 \times (3)^2 \times 8
\]
\[
V = 3.142 \times 9 \times 8
\]
\[
V = 3.142 \times 72
\]
\[
V = 226.224 \, \text{cm}^3
\]

Thus, the volume of the cylinder is **226.224 cm³**.

#### 4.2.4: Show that total surface area is given by \( 2\pi r^2 + 2\pi rh \)
The total surface area \( SA \) of a cylinder is the sum of the areas of the two circular faces and the lateral surface area. The formula is:
\[
SA = 2\pi r^2 + 2\pi rh
\]
Where:
- \( 2\pi r^2 \) is the area of the two circular faces.
- \( 2\pi rh \) is the lateral surface area.

#### 4.2.5: Calculate the total surface area
Now, use the formula for surface area:
\[
SA = 2\pi r^2 + 2\pi rh
\]
Substitute the values:
\[
SA = 2 \times 3.142 \times (3)^2 + 2 \times 3.142 \times 3 \times 8
\]
\[
SA = 2 \times 3.142 \times 9 + 2 \times 3.142 \times 24
\]
\[
SA = 56.556 + 150.816
\]
\[
SA = 207.372 \, \text{cm}^2
\]

Thus, the total surface area of the cylinder is **207.372 cm²**.

---

Would you like further details on any of these steps or additional clarifications?

### Related Questions:
1. How would the volume change if the height of the rectangular prism doubled?
2. What would be the volume of the cylinder if the radius increased by 50%?
3. How do the formulas for volume differ between prisms and cylinders?
4. What is the relationship between the radius and surface area in a cylinder?
5. How would the surface area of the prism change if the height was increased by 2 cm?

**Tip**: Remember that in many geometry problems, understanding the formulas for volume and surface area is key to solving for unknowns like height or radius.

4.1 The rectangular prism above has a volume of 270 cm³. Calculate the value of the height. 4.2 4.2.1 Identify the shape above. 4.2.2 How many faces does it have? 4.2.3 Calculate the volume of the shape if r = 3 cm, h = 8 cm, π = 3.142. 4.2.4 Show that total surface area is given by 2πr² + 2πrh. 4.2.5 Calculate the total surface area of the shape.

Learn how to calculate the volume of a rectangular prism with given dimensions and the height of the prism. Also, explore the surface area and volume of a cylinder, with a step-by-step solution for Grade 10 mathematics.

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