Question

Right Rectangular Prisms & Pyramids Unit Test

Answer 1

Sure! To help you prepare for a unit test on right rectangular prisms and pyramids, I can provide you with some sample questions and concepts that you might need to know.

### Key Concepts

1. **Definitions:**
- **Right Rectangular Prism:** A three-dimensional figure with six rectangular faces, where all angles are right angles. It has length, width, and height.
- **Pyramid:** A three-dimensional figure that has a polygonal base and triangular faces that meet at a common point (the apex).

2. **Volume Formulas:**
- **Volume of a Right Rectangular Prism:** \( V = l \times w \times h \)
- where \( l \) is length, \( w \) is width, and \( h \) is height.
- **Volume of a Pyramid:** \( V = \frac{1}{3} \times B \times h \)
- where \( B \) is the area of the base, and \( h \) is the height from the base to the apex.

3. **Surface Area Formulas:**
- **Surface Area of a Right Rectangular Prism:** \( SA = 2(lw + lh + wh) \)
- **Surface Area of a Pyramid:** The surface area is the sum of the base area and the areas of the triangular faces.
- For a pyramid with a square base: \( SA = B + \frac{1}{2} \times P \times l \)
- where \( B \) is the area of the base, \( P \) is the perimeter of the base, and \( l \) is the slant height.

### Sample Questions

1. **Calculate Volume:**
- A right rectangular prism has a length of 5 cm, a width of 3 cm, and a height of 4 cm. What is its volume?

2. **Calculate Surface Area:**
- Determine the surface area of a right rectangular prism with dimensions 4 cm x 7 cm x 3 cm.

3. **Pyramid Volume Calculation:**
- A pyramid has a square base with a side length of 6 m and a height of 9 m. What is the volume of the pyramid?

4. **Pyramid Surface Area Calculation:**
- Calculate the surface area of a triangular pyramid with a base area of 12 m² and a slant height of 5 m. Assume that the perimeter of the base is 16 m.

5. **Comparison Problem:**
- If a rectangular prism has a volume of 120 cm³ and a height of 5 cm, what are the possible dimensions for the length and width?

6. **Application Problem:**
- You are designing a box that must hold a volume of 1000 cm³. If the box is a right rectangular prism and has a height of 10 cm, what dimensions should it have?

### Answers

1. For the first question:
\[
V = 5 \times 3 \times 4 = 60 \, \text{cm}^3
\]

2. For the surface area:
\[
SA = 2(4 \times 7 + 4 \times 3 + 7 \times 3) = 2(28 + 12 + 21) = 2 \times 61 = 122 \, \text{cm}^2
\]

3. For the pyramid:
\[
V = \frac{1}{3} \times (6 \times 6) \times 9 = \frac{1}{3} \times 36 \times 9 = 108 \, \text{m}^3
\]

4. For the surface area of the pyramid:
\[
SA = 12 + \frac{1}{2} \times 16 \times 5 = 12 + 40 = 52 \, \text{m}^2
\]

5. For the comparison problem, using volume:
\[
120 = 5 \times l \times w \implies l \times w = \frac{120}{5} = 24 \quad \text{(possible pairs include (1, 24), (2, 12), (3, 8), (4, 6))}
\]

6. For the application problem:
\[
V = l \times w \times 10 = 1000 \implies l \times w = 100 \quad \text{(possible pairs include (1, 100), (2, 50), (4, 25), (5, 20), (10, 10))}
\]

Feel free to modify these questions or ask for more specific topics or examples if needed! Good luck with your unit test!