Writing an Expression for the Height of Each Post

Oct 13, 2025 by talynt.org 11 views

Introduction

When it comes to designing a guardrail for the steps leading up to your home, safety is the top priority. The guardrail should be sturdy enough to prevent falls, yet not so tall that it obstructs the view or creates a sense of confinement. In this article, we will explore the mathematical concept of a guardrail with multiple posts, each with a specific height. We will use the given information to write an expression for the height of each post, starting from the tallest one.

The Problem

We are given that there are 5 posts supporting the guardrail, with the tallest post being 3 feet tall. The height of each of the other posts is $\frac{5}{6}$ the height of the next larger post. This means that the second tallest post is $\frac{5}{6}$ times the height of the tallest post, the third tallest post is $\frac{5}{6}$ times the height of the second tallest post, and so on.

Writing an Expression for the Height of Each Post

Let's denote the height of the tallest post as $h_1$ . Since the second tallest post is $\frac{5}{6}$ times the height of the tallest post, its height can be expressed as $\frac{5}{6}h_1$ . Similarly, the height of the third tallest post is $\frac{5}{6}$ times the height of the second tallest post, which can be expressed as $\frac{5}{6}\left(\frac{5}{6}h_1\right)$ . We can continue this pattern to find the height of each post.

Finding the Height of Each Post

Let's start with the tallest post, which has a height of 3 feet. We can express its height as $h_1 = 3$ .

The second tallest post has a height of $\frac{5}{6}h_1$ , which can be expressed as:

h_2 = \frac{5}{6}h_1 = \frac{5}{6}(3) = \frac{15}{6} = 2.5

The third tallest post has a height of $\frac{5}{6}h_2$ , which can be expressed as:

h_3 = \frac{5}{6}h_2 = \frac{5}{6}\left(\frac{15}{6}\right) = \frac{75}{36} = 2.0833

The fourth tallest post has a height of $\frac{5}{6}h_3$ , which can be expressed as:

h_4 = \frac{5}{6}h_3 = \frac{5}{6}\left(\frac{75}{36}\right) = \frac{375}{216} = 1.7361

The fifth and shortest post has a height of $\frac{5}{6}h_4$ , which can be expressed as:

h_5 = \frac{5}{6}h_4 = \frac{5}{6}\left(\frac{375}{216}\right) = \frac{1875}{1296} = 1.4472

Conclusion

In this article, we used the given information to write an expression for the height of each post in a guardrail with multiple posts. We started with the tallest post, which has a height of 3 feet, and used the given ratio to find the height of each subsequent post. The resulting expressions for the height of each post are:

h_1 = 3

h_2 = \frac{15}{6} = 2.5

h_3 = \frac{75}{36} = 2.0833

h_4 = \frac{375}{216} = 1.7361

h_5 = \frac{1875}{1296} = 1.4472

These expressions can be used to design a guardrail with multiple posts, each with a specific height, while ensuring that the guardrail is sturdy enough to prevent falls and does not obstruct the view or create a sense of confinement.

Introduction

In our previous article, we explored the mathematical concept of a guardrail with multiple posts, each with a specific height. We used the given information to write an expression for the height of each post, starting from the tallest one. In this article, we will answer some frequently asked questions related to the guardrail design and provide additional insights into the mathematical concept.

Q&A

Q: What is the purpose of a guardrail?

A: A guardrail is a safety feature designed to prevent falls and injuries. It is typically installed at the edge of a staircase, balcony, or other elevated area to prevent people from falling off.

Q: Why is it important to have multiple posts in a guardrail?

A: Having multiple posts in a guardrail provides additional support and stability, which is essential for preventing falls and injuries. The multiple posts also help to distribute the weight of the guardrail evenly, reducing the risk of collapse.

Q: How do I determine the height of each post in a guardrail?

A: To determine the height of each post in a guardrail, you can use the formula we derived in our previous article:

h_n = \frac{5}{6}h_{n-1}

where $h_n$ is the height of the $n$ th post, and $h_{n-1}$ is the height of the $(n-1)$ th post.

Q: What is the significance of the ratio $\frac{5}{6}$ in the guardrail design?

A: The ratio $\frac{5}{6}$ is used to determine the height of each post in the guardrail. This ratio ensures that each post is slightly shorter than the previous one, providing a gradual decrease in height as you move from the top to the bottom of the guardrail.

Q: Can I use a different ratio to design a guardrail?

A: Yes, you can use a different ratio to design a guardrail. However, the ratio $\frac{5}{6}$ is commonly used in guardrail design because it provides a good balance between safety and aesthetics.

Q: How do I ensure that the guardrail is sturdy enough to prevent falls?

A: To ensure that the guardrail is sturdy enough to prevent falls, you should use a combination of strong materials and a robust design. The guardrail should be able to withstand the weight of people and any external forces, such as wind or earthquakes.

Q: Can I use a guardrail with multiple posts in a residential setting?

A: Yes, you can use a guardrail with multiple posts in a residential setting. However, you should ensure that the guardrail is designed and installed according to local building codes and regulations.

Conclusion

In this article, we answered some frequently asked questions related to the guardrail design and provided additional insights into the mathematical concept. We hope that this article has been helpful in understanding the importance of guardrails and how to design them using mathematical principles.

Additional Resources

If you have any further questions or would like to learn more about guardrail design, we recommend the following resources:

American Society of Civil Engineers (ASCE): The ASCE provides guidelines and standards for guardrail design and installation.
International Building Code (IBC): The IBC provides regulations and guidelines for guardrail design and installation in residential and commercial settings.
National Institute for Occupational Safety and Health (NIOSH): NIOSH provides guidelines and recommendations for guardrail design and installation in workplaces.

We hope that this article has been helpful in understanding the importance of guardrails and how to design them using mathematical principles. If you have any further questions or would like to learn more, please don't hesitate to contact us.