Finding the Length of the Box

Oct 13, 2025 by pynkocean.com 12 views

Introduction

In this problem, we are tasked with finding the equation that represents the volume of a box with a rectangular base. The height of the box is given as 10 inches, and the perimeter of the rectangular base is 72 inches. We will use the variables ${$ v $}$ to represent the volume of the box and ${$ w $}$ to represent the width of the box.

Given Information

Height of the box ( ${$ h $}$): 10 inches
Perimeter of the rectangular base: 72 inches

Formulating the Equation

To find the equation that represents the volume of the box, we need to use the formula for the volume of a rectangular prism, which is given by:

${$ v = lwh $}$

where ${$ l $}$ is the length, ${$ w $}$ is the width, and ${$ h $}$ is the height.

However, we are not given the length of the box. Instead, we are given the perimeter of the rectangular base, which is 72 inches. The perimeter of a rectangle is given by:

${$ P = 2l + 2w $}$

We can use this equation to find the length of the box in terms of the width.

Finding the Length of the Box

We are given that the perimeter of the rectangular base is 72 inches. We can set up an equation using the formula for the perimeter of a rectangle:

${$ 72 = 2l + 2w $}$

We can simplify this equation by dividing both sides by 2:

${$ 36 = l + w $}$

Now, we can solve for the length of the box in terms of the width:

${$ l = 36 - w $}$

Substituting the Length into the Volume Equation

Now that we have found the length of the box in terms of the width, we can substitute this expression into the volume equation:

${$ v = (36 - w)wh $}$

We can simplify this equation by distributing the width to the terms inside the parentheses:

${$ v = 36wh - wh^2 $}$

However, we are given that the height of the box is 10 inches. We can substitute this value into the equation:

${$ v = 36w(10) - w(10)^2 $}$

Simplifying this equation, we get:

${$ v = 360w - 100w $}$

Combining like terms, we get:

${$ v = 260w $}$

Conclusion

In this problem, we were tasked with finding the equation that represents the volume of a box with a rectangular base. We were given the height of the box and the perimeter of the rectangular base. We used the formula for the perimeter of a rectangle to find the length of the box in terms of the width, and then substituted this expression into the volume equation. The resulting equation is:

${$ v = 260w $}$

This equation represents the volume of the box in terms of the width of the box.

Final Answer

Q&A: Understanding the Problem and Solution

Q: What is the problem asking us to find?

A: The problem is asking us to find the equation that represents the volume of a box with a rectangular base. The height of the box is given as 10 inches, and the perimeter of the rectangular base is 72 inches.

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 = lwh $}$

where ${$ l $}$ is the length, ${$ w $}$ is the width, and ${$ h $}$ is the height.

Q: How do we find the length of the box?

A: We are given the perimeter of the rectangular base, which is 72 inches. We can use the formula for the perimeter of a rectangle to find the length of the box in terms of the width:

${$ 72 = 2l + 2w $}$

Simplifying this equation, we get:

${$ 36 = l + w $}$

Now, we can solve for the length of the box in terms of the width:

${$ l = 36 - w $}$

Q: How do we substitute the length into the volume equation?

A: We can substitute the expression for the length into the volume equation:

${$ v = (36 - w)wh $}$

Simplifying this equation, we get:

${$ v = 36wh - wh^2 $}$

Substituting the value of the height, we get:

${$ v = 36w(10) - w(10)^2 $}$

Simplifying this equation, we get:

${$ v = 360w - 100w $}$

Combining like terms, we get:

${$ v = 260w $}$

Q: What is the final equation that represents the volume of the box?

A: The final equation that represents the volume of the box is:

${$ v = 260w $}$

This equation represents the volume of the box in terms of the width of the box.

Q: What is the significance of the equation?

A: The equation ${$ v = 260w $}$ represents the volume of the box in terms of the width of the box. This equation can be used to calculate the volume of the box for any given width.

Q: What are some real-world applications of this problem?

A: This problem has real-world applications in architecture, engineering, and design. For example, architects and engineers use mathematical equations to design buildings and structures that are safe and efficient. Designers use mathematical equations to create models and prototypes of products and buildings.

Q: How can we use this problem to learn more about mathematics?

A: This problem can be used to learn more about mathematics by exploring the concepts of volume, perimeter, and area. It can also be used to learn about algebraic equations and how to solve them. Additionally, it can be used to learn about real-world applications of mathematics and how it is used in various fields.

Conclusion

In this article, we explored the problem of finding the equation that represents the volume of a box with a rectangular base. We used the formula for the perimeter of a rectangle to find the length of the box in terms of the width, and then substituted this expression into the volume equation. The resulting equation is:

${$ v = 260w $}$

This equation represents the volume of the box in terms of the width of the box. We also explored the significance of the equation and its real-world applications.