The Volume of a Rectangular Block

Oct 13, 2025 by wbrum.org 140 views

Introduction

In mathematics, the volume of a rectangular block is a fundamental concept that is used to calculate the amount of space inside the block. In this article, we will explore the volume of a rectangular block of clay, given by the expression $(2x-1)(x-3)(x+4)$ . We will examine the properties of this expression and determine which statement about the volume of the block of clay is true.

The Volume of a Rectangular Block

The volume of a rectangular block is given by the product of its length, width, and height. In the case of the block of clay, we are given the expression $(2x-1)(x-3)(x+4)$ , which represents the volume of the block. To understand the properties of this expression, we need to expand it and examine its factors.

Expanding the Expression

To expand the expression $(2x-1)(x-3)(x+4)$ , we can use the distributive property of multiplication over addition. This means that we can multiply each term in the first factor by each term in the second factor, and then multiply the result by the third factor.

(2x-1)(x-3)(x+4) = (2x-1)(x^2 - 3x + 4x - 12)(x+4)

Expanding the expression further, we get:

(2x-1)(x^2 - 3x + 4x - 12)(x+4) = (2x-1)(x^2 + x - 12)(x+4)

Now, we can multiply the first two factors:

(2x-1)(x^2 + x - 12) = 2x^3 + 2x^2 - 24x - x^2 - x + 12

Combining like terms, we get:

2x^3 + x^2 - 25x + 12

Now, we can multiply the result by the third factor:

(2x^3 + x^2 - 25x + 12)(x+4) = 2x^4 + 8x^3 + x^3 + 4x^2 - 25x^2 - 100x + 12x + 48

Combining like terms, we get:

2x^4 + 9x^3 - 24x^2 - 88x + 48

The Factors of the Expression

Now that we have expanded the expression, we can examine its factors. The expression can be factored as:

(2x-1)(x-3)(x+4) = (2x-1)(x^2 + x - 12)(x+4)

This means that the expression has three factors: $(2x-1)$ , $(x^2 + x - 12)$ , and $(x+4)$ . Each of these factors represents a different aspect of the volume of the block of clay.

The Area of the Base

The area of the base of the block of clay is given by the expression $(x-3)(x+4)$ . This is because the base of the block is a rectangle with length $(x-3)$ and width $(x+4)$ .

The Height of the Block

The height of the block of clay is given by the expression $(2x-1)$ . This is because the height of the block is a rectangle with length $(2x-1)$ and width $(x-3)$ .

The Volume of the Block

The volume of the block of clay is given by the expression $(2x-1)(x-3)(x+4)$ . This is because the volume of the block is the product of its length, width, and height.

Conclusion

In conclusion, the volume of the block of clay is given by the expression $(2x-1)(x-3)(x+4)$ . This expression represents the product of the length, width, and height of the block. The area of the base of the block is given by the expression $(x-3)(x+4)$ , and the height of the block is given by the expression $(2x-1)$ . Therefore, the statement that the volume is the product of the area of the base and the height of the block is true.

Answer

The correct answer is:

A. The volume is the product of the area of the base and the height of the block.

Discussion

This problem requires the student to understand the concept of volume and how it is related to the area of the base and the height of the block. The student must be able to expand the expression and examine its factors in order to determine the correct answer.

Key Concepts

The volume of a rectangular block is given by the product of its length, width, and height.
The area of the base of the block is given by the expression $(x-3)(x+4)$ .
The height of the block is given by the expression $(2x-1)$ .
The volume of the block is given by the expression $(2x-1)(x-3)(x+4)$ .

Practice Problems

Find the volume of a rectangular block with length $(x-2)$ , width $(x+3)$ , and height $(x+1)$ .
Find the area of the base of a rectangular block with length $(x-1)$ and width $(x+2)$ .
Find the height of a rectangular block with length $(x+2)$ and width $(x-1)$ .

Solutions

The volume of the block is given by the expression $(x-2)(x+3)(x+1)$ .
The area of the base of the block is given by the expression $(x-1)(x+2)$ .
The height of the block is given by the expression $(x+2)$ .

Conclusion

Introduction

In our previous article, we explored the volume of a rectangular block of clay, given by the expression $(2x-1)(x-3)(x+4)$ . We examined the properties of this expression and determined which statement about the volume of the block of clay is true. In this article, we will answer some frequently asked questions about the volume of a rectangular block of clay.

Q: What is the volume of a rectangular block of clay?

A: The volume of a rectangular block of clay is given by the expression $(2x-1)(x-3)(x+4)$ .

Q: How do I calculate the volume of a rectangular block of clay?

A: To calculate the volume of a rectangular block of clay, you need to multiply the length, width, and height of the block. The length, width, and height of the block are given by the expressions $(2x-1)$ , $(x-3)$ , and $(x+4)$ , respectively.

Q: What is the area of the base of a rectangular block of clay?

A: The area of the base of a rectangular block of clay is given by the expression $(x-3)(x+4)$ .

Q: What is the height of a rectangular block of clay?

A: The height of a rectangular block of clay is given by the expression $(2x-1)$ .

Q: How do I find the volume of a rectangular block of clay with a given length, width, and height?

A: To find the volume of a rectangular block of clay with a given length, width, and height, you need to multiply the length, width, and height of the block. For example, if the length of the block is $(x-2)$ , the width is $(x+3)$ , and the height is $(x+1)$ , the volume of the block is given by the expression $(x-2)(x+3)(x+1)$ .

Q: What is the relationship between the volume of a rectangular block of clay and its dimensions?

A: The volume of a rectangular block of clay is directly proportional to its dimensions. This means that if the length, width, and height of the block are increased or decreased, the volume of the block will also increase or decrease proportionally.

Q: Can I use the formula for the volume of a rectangular block of clay to find the volume of a cube?

A: Yes, you can use the formula for the volume of a rectangular block of clay to find the volume of a cube. A cube is a special type of rectangular block with equal length, width, and height. If the length, width, and height of the cube are all equal to $(x)$ , the volume of the cube is given by the expression $(x)^3$ .

Q: How do I apply the formula for the volume of a rectangular block of clay to real-world problems?

A: The formula for the volume of a rectangular block of clay can be applied to a wide range of real-world problems, such as calculating the volume of a box, a container, or a building. For example, if you are designing a box with a length of $(x-2)$ , a width of $(x+3)$ , and a height of $(x+1)$ , you can use the formula to calculate the volume of the box.

Conclusion

In conclusion, the volume of a rectangular block of clay is given by the expression $(2x-1)(x-3)(x+4)$ . The area of the base of the block is given by the expression $(x-3)(x+4)$ , and the height of the block is given by the expression $(2x-1)$ . We hope that this article has helped to clarify any questions you may have had about the volume of a rectangular block of clay. If you have any further questions, please don't hesitate to ask.

Practice Problems

Find the volume of a rectangular block with length $(x-2)$ , width $(x+3)$ , and height $(x+1)$ .
Find the area of the base of a rectangular block with length $(x-1)$ and width $(x+2)$ .
Find the height of a rectangular block with length $(x+2)$ and width $(x-1)$ .

Solutions

The volume of the block is given by the expression $(x-2)(x+3)(x+1)$ .
The area of the base of the block is given by the expression $(x-1)(x+2)$ .
The height of the block is given by the expression $(x+2)$ .

Key Concepts

The volume of a rectangular block of clay is given by the expression $(2x-1)(x-3)(x+4)$ .
The area of the base of the block is given by the expression $(x-3)(x+4)$ .
The height of the block is given by the expression $(2x-1)$ .
The volume of a rectangular block of clay is directly proportional to its dimensions.