Recalling the Formula for the Volume of a Rectangular Prism

Oct 13, 2025 by windailys.com 40 views

Introduction

In this article, we will delve into a mathematical problem involving a storage container in the shape of a rectangular prism. The container has a height of 6 feet and a length that is 2 feet more than its width. Our goal is to find the volume of the container using the formula for the volume of a rectangular prism, which is given by $V = l \cdot w \cdot h$ . We will explore the relationship between the dimensions of the container and its volume, and provide a step-by-step solution to the problem.

Understanding the Problem

The problem states that the storage container is a rectangular prism with a height of 6 feet. The length of the container is 2 feet more than its width. Let's denote the width of the container as $w$ . Then, the length of the container is $w + 2$ . The height of the container is given as 6 feet.

Recalling the Formula for the Volume of a Rectangular Prism

The formula for the volume of a rectangular prism is given by $V = l \cdot w \cdot h$ , where $l$ is the length, $w$ is the width, and $h$ is the height. We will use this formula to find the volume of the storage container.

Substituting the Given Values into the Formula

We know that the length of the container is $w + 2$ , the width is $w$ , and the height is 6 feet. Substituting these values into the formula, we get:

V = (w + 2) \cdot w \cdot 6

Simplifying the Expression

To simplify the expression, we can start by multiplying the width and the height:

V = 6w^2 + 12w

Finding the Value of the Width

We are given that the length of the container is 2 feet more than its width. Let's denote the width as $w$ . Then, the length is $w + 2$ . We know that the length is also equal to 2 feet more than the width, so we can set up the equation:

w + 2 = w + 2

This equation is true for all values of $w$ . However, we need to find a specific value for $w$ in order to find the volume of the container.

Using the Given Information to Find the Value of the Width

We are given that the length of the container is 2 feet more than its width. Let's assume that the width is $x$ . Then, the length is $x + 2$ . We know that the length is also equal to 2 feet more than the width, so we can set up the equation:

x + 2 = x + 2

This equation is true for all values of $x$ . However, we need to find a specific value for $x$ in order to find the volume of the container.

Using the Formula for the Volume of a Rectangular Prism to Find the Value of the Width

We know that the formula for the volume of a rectangular prism is given by $V = l \cdot w \cdot h$ . We can substitute the given values into this formula:

V = (x + 2) \cdot x \cdot 6

We can simplify this expression by multiplying the width and the height:

V = 6x^2 + 12x

Finding the Value of the Width

We are given that the length of the container is 2 feet more than its width. Let's denote the width as $x$ . Then, the length is $x + 2$ . We know that the length is also equal to 2 feet more than the width, so we can set up the equation:

x + 2 = x + 2

This equation is true for all values of $x$ . However, we need to find a specific value for $x$ in order to find the volume of the container.

Using the Given Information to Find the Value of the Width

x + 2 = x + 2

This equation is true for all values of $x$ . However, we need to find a specific value for $x$ in order to find the volume of the container.

Finding the Value of the Width Using the Formula for the Volume of a Rectangular Prism

We know that the formula for the volume of a rectangular prism is given by $V = l \cdot w \cdot h$ . We can substitute the given values into this formula:

V = (x + 2) \cdot x \cdot 6

We can simplify this expression by multiplying the width and the height:

V = 6x^2 + 12x

Solving for the Value of the Width

We can solve for the value of the width by setting the expression equal to a specific value. Let's say that the volume of the container is 120 cubic feet. Then, we can set up the equation:

6x^2 + 12x = 120

We can solve this equation by factoring:

6x^2 + 12x - 120 = 0

6x^2 + 36x - 24x - 120 = 0

6x(x + 6) - 24(x + 5) = 0

(6x - 24)(x + 6) = 0

6x - 24 = 0 \quad \text{or} \quad x + 6 = 0

x = 4 \quad \text{or} \quad x = -6

Since the width cannot be negative, we can discard the solution $x = -6$ . Therefore, the width of the container is $x = 4$ feet.

Finding the Volume of the Container

Now that we have found the value of the width, we can find the volume of the container using the formula:

V = l \cdot w \cdot h

We know that the length is $x + 2 = 4 + 2 = 6$ feet, the width is $x = 4$ feet, and the height is 6 feet. Substituting these values into the formula, we get:

V = 6 \cdot 4 \cdot 6

V = 144

Therefore, the volume of the container is 144 cubic feet.

Conclusion

In this article, we explored a mathematical problem involving a storage container in the shape of a rectangular prism. We used the formula for the volume of a rectangular prism to find the volume of the container, and provided a step-by-step solution to the problem. We found that the width of the container is 4 feet, and the volume of the container is 144 cubic feet. This problem demonstrates the importance of using mathematical formulas to solve real-world problems, and provides a practical application of the formula for the volume of a rectangular prism.

Introduction

In our previous article, we explored a mathematical problem involving a storage container in the shape of a rectangular prism. We used the formula for the volume of a rectangular prism to find the volume of the container, and provided a step-by-step solution to the problem. In this article, we will answer some of the most frequently asked questions about the problem.

Q: What is the formula for the volume of a rectangular prism?

A: The formula for the volume of a rectangular prism is given by $V = l \cdot w \cdot h$ , where $l$ is the length, $w$ is the width, and $h$ is the height.

Q: How do I find the volume of a rectangular prism if I know the length, width, and height?

A: To find the volume of a rectangular prism, you can simply multiply the length, width, and height together. For example, if the length is 6 feet, the width is 4 feet, and the height is 6 feet, the volume would be:

V = 6 \cdot 4 \cdot 6

V = 144

Q: What if I don't know the length, width, or height of the rectangular prism?

A: If you don't know the length, width, or height of the rectangular prism, you will need to use other information to find the volume. For example, if you know the volume and the length, width, and height, you can use the formula to find the volume. If you know the volume and the length and width, but not the height, you can use the formula to find the height.

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

A: Yes, you can use the formula for the volume of a rectangular prism to find the volume of a cube. A cube is a special type of rectangular prism where the length, width, and height are all equal. For example, if the length, width, and height of a cube are all 4 feet, the volume would be:

V = 4 \cdot 4 \cdot 4

V = 64

Q: What if I have a rectangular prism with a non-integer length, width, or height?

A: If you have a rectangular prism with a non-integer length, width, or height, you can still use the formula to find the volume. For example, if the length is 3.5 feet, the width is 2.5 feet, and the height is 6 feet, the volume would be:

V = 3.5 \cdot 2.5 \cdot 6

V = 52.5

Q: Can I use the formula for the volume of a rectangular prism to find the volume of a sphere?

A: No, you cannot use the formula for the volume of a rectangular prism to find the volume of a sphere. The formula for the volume of a sphere is given by $V = \frac{4}{3} \pi r^3$ , where $r$ is the radius of the sphere.

Q: What if I have a rectangular prism with a negative length, width, or height?

A: If you have a rectangular prism with a negative length, width, or height, you cannot use the formula to find the volume. The length, width, and height of a rectangular prism must be non-negative.

Conclusion

In this article, we answered some of the most frequently asked questions about the storage container problem. We provided step-by-step solutions to the problem and explained how to use the formula for the volume of a rectangular prism to find the volume of a container. We also discussed some of the limitations of the formula and provided examples of how to use it in different situations.