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answer all of these questions correctly with clear working out

The image you've uploaded contains a set of problems related to 3D shapes and their properties. I'll go through each problem, solve it, and provide clear working out.

### 1. Surface Area of a Rectangular Prism

The surface area of a rectangular prism is calculated using the formula:
\[
\text{Surface Area} = 2lw + 2lh + 2wh
\]
where \( l \) is the length, \( w \) is the width, and \( h \) is the height.

Let's denote:
- \( l = 10 \) units
- \( w = 4 \) units
- \( h = 5 \) units

Substituting these values:
\[
\text{Surface Area} = 2(10)(4) + 2(10)(5) + 2(4)(5) = 80 + 100 + 40 = 220 \text{ square units}
\]

### 2. Volume of a Triangular Prism

The volume of a triangular prism is calculated using the formula:
\[
\text{Volume} = \text{Base Area} \times \text{Height}
\]
For a triangular base:
\[
\text{Base Area} = \frac{1}{2} \times \text{Base} \times \text{Height of Triangle}
\]
Given:
- Base of triangle = 6 units
- Height of triangle = 4 units
- Length/Height of prism = 8 units

\[
\text{Base Area} = \frac{1}{2} \times 6 \times 4 = 12 \text{ square units}
\]
\[
\text{Volume} = 12 \times 8 = 96 \text{ cubic units}
\]

### 3. Surface Area of a Cylinder

The surface area of a cylinder is calculated using the formula:
\[
\text{Surface Area} = 2\pi rh + 2\pi r^2
\]
where \( r \) is the radius and \( h \) is the height.

Given:
- \( r = 3 \) units
- \( h = 10 \) units

Substituting these into the formula:
\[
\text{Surface Area} = 2\pi (3)(10) + 2\pi (3)^2 = 60\pi + 18\pi = 78\pi \text{ square units}
\]

### 4. Volume of a Pyramid

The volume of a pyramid is calculated using the formula:
\[
\text{Volume} = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
Given:
- Base area = 20 square units
- Height = 9 units

Substituting:
\[
\text{Volume} = \frac{1}{3} \times 20 \times 9 = 60 \text{ cubic units}
\]

### 5. Volume of a Cone

The volume of a cone is calculated using the formula:
\[
\text{Volume} = \frac{1}{3} \pi r^2 h
\]
Given:
- \( r = 4 \) units
- \( h = 6 \) units

Substituting:
\[
\text{Volume} = \frac{1}{3} \pi (4)^2(6) = \frac{1}{3} \pi (16)(6) = \frac{1}{3} \pi 96 = 32\pi \text{ cubic units}
\]

These are the solutions to the problems on the image. If you need more details or further questions, feel free to ask!

Here are five related questions:
1. How would the surface area change if the height of the rectangular prism were doubled?
2. What is the volume of a triangular prism if the base area is halved?
3. How does the surface area of a cylinder change with respect to its radius?
4. Can you derive the surface area formula for a cone?
5. How does the volume of a pyramid compare to a prism with the same base area and height?

**Tip:** Understanding how formulas for 3D shapes are derived helps in remembering them and applying them correctly!

Learn how to calculate surface area and volume for 3D shapes like rectangular prisms, triangular prisms, cylinders, pyramids, and cones. Detailed solutions and step-by-step calculations provided.

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