Representing the Height with a Function

Oct 13, 2025 by miodeya.com 132 views

Introduction

When designing a railing for a set of stairs, it's essential to consider the height and slant of the stairs to ensure a safe and aesthetically pleasing design. In this article, we'll explore how to represent the height of the railing using a mathematical function.

Understanding the Problem

Caitlin is designing a railing for a set of stairs that begins at a height of 36 inches. The railing follows the slant of the stairs, which decreases 9 inches for every 12 horizontal inches. This means that for every 12 inches of horizontal distance, the height of the railing decreases by 9 inches.

Representing the Height with a Function

To represent the height of the railing, we can use a linear function. A linear function is a function that can be written in the form:

y = mx + b

where:

y is the output or dependent variable
x is the input or independent variable
m is the slope of the line
b is the y-intercept

In this case, we want to represent the height of the railing (y) as a function of the horizontal distance (x). We know that the railing begins at a height of 36 inches, so we can set the y-intercept (b) to 36.

Finding the Slope

To find the slope (m), we need to determine how much the height of the railing changes for every 12 inches of horizontal distance. We know that the height decreases by 9 inches for every 12 inches of horizontal distance, so the slope is:

m = -9/12 = -0.75

The negative sign indicates that the height of the railing decreases as the horizontal distance increases.

Writing the Function

Now that we have the slope and y-intercept, we can write the function that represents the height of the railing:

y = -0.75x + 36

This function tells us that for every 12 inches of horizontal distance, the height of the railing decreases by 9 inches, starting from a height of 36 inches.

Interpreting the Function

To interpret this function, let's consider a few examples:

If the horizontal distance is 0 inches, the height of the railing is 36 inches (y = -0.75(0) + 36 = 36).
If the horizontal distance is 12 inches, the height of the railing is 27 inches (y = -0.75(12) + 36 = 27).
If the horizontal distance is 24 inches, the height of the railing is 18 inches (y = -0.75(24) + 36 = 18).

Conclusion

In this article, we explored how to represent the height of a railing for a set of stairs using a mathematical function. We used a linear function to model the height of the railing, taking into account the slope and y-intercept. By understanding the function that represents the height of the railing, designers like Caitlin can create safe and aesthetically pleasing designs for their stair railings.

Mathematical Representation

The function that represents the height of the railing is:

y = -0.75x + 36

where:

y is the height of the railing
x is the horizontal distance
-0.75 is the slope of the line
36 is the y-intercept

Graphical Representation

The graph of the function is a straight line with a negative slope. The line passes through the point (0, 36) and has a slope of -0.75.

Real-World Applications

The mathematical representation of the height of a railing has real-world applications in various fields, including:

Architecture: Designers use mathematical functions to create safe and aesthetically pleasing designs for stair railings.
Engineering: Engineers use mathematical functions to model the behavior of complex systems, including stair railings.
Science: Scientists use mathematical functions to analyze data and make predictions about the behavior of complex systems.

Future Directions

In the future, researchers may explore new mathematical representations for the height of stair railings, taking into account factors such as:

Non-linear relationships between height and horizontal distance
Variations in the slope of the line
Interactions between the railing and other components of the staircase

By exploring these new mathematical representations, researchers can create even more accurate and effective designs for stair railings.