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!