Understanding the Side Length Function

Oct 13, 2025 by wbrum.org 91 views

Introduction

In this article, we will delve into the world of mathematics and explore the concept of surface area, specifically in the context of an ice cube. We will examine how the side length of the ice cube changes over time, and how this affects its surface area. The side length of the ice cube is given by the function ${$ s(t) = \frac{1}{2} t + 2$}$, where ${$ t$}$ is in hours. We will also discuss the function ${$ A(s) = 6s^2$}$, which describes the surface area of the ice cube in terms of its side length.

Understanding the Side Length Function

The side length of the ice cube is given by the function ${$ s(t) = \frac{1}{2} t + 2$}$. This function indicates that the side length of the ice cube increases at a rate of ${$ \frac{1}{2}$}$ inch per hour. In other words, for every hour that passes, the side length of the ice cube increases by ${$ \frac{1}{2}$}$ inch.

To better understand this function, let's consider a few examples. If ${$ t = 0$}$, then the side length of the ice cube is ${$ s(0) = \frac{1}{2} (0) + 2 = 2$}$ inches. This makes sense, as the ice cube is initially a cube with a side length of 2 inches. If ${$ t = 1$}$, then the side length of the ice cube is ${$ s(1) = \frac{1}{2} (1) + 2 = 2.5$}$ inches. This indicates that after 1 hour, the side length of the ice cube has increased by ${$ \frac{1}{2}$}$ inch.

Understanding the Surface Area Function

The surface area of the ice cube is described by the function ${$ A(s) = 6s^2$}$. This function indicates that the surface area of the ice cube is directly proportional to the square of its side length. In other words, as the side length of the ice cube increases, its surface area increases at a much faster rate.

To better understand this function, let's consider a few examples. If ${$ s = 2$}$ inches, then the surface area of the ice cube is ${$ A(2) = 6(2)^2 = 24$}$ square inches. This makes sense, as the surface area of a cube with a side length of 2 inches is indeed 24 square inches. If ${$ s = 2.5$}$ inches, then the surface area of the ice cube is ${$ A(2.5) = 6(2.5)^2 = 56.25$}$ square inches. This indicates that as the side length of the ice cube increases, its surface area increases at a much faster rate.

Combining the Two Functions

Now that we have a good understanding of both the side length function and the surface area function, let's combine them to see how the surface area of the ice cube changes over time. We can do this by substituting the side length function into the surface area function.

Substituting ${$ s(t) = \frac{1}{2} t + 2$}$ into ${$ A(s) = 6s^2$}$, we get:

${$ A(t) = 6\left(\frac{1}{2} t + 2\right)^2$}$

Expanding this expression, we get:

${$ A(t) = 6\left(\frac{1}{4} t^2 + 2t + 4\right)$}$

Simplifying this expression, we get:

${$ A(t) = \frac{3}{2} t^2 + 12t + 24$}$

This function describes the surface area of the ice cube in terms of time. We can use this function to determine the surface area of the ice cube at any given time.

Graphing the Surface Area Function

To get a better understanding of how the surface area of the ice cube changes over time, let's graph the surface area function.

The graph of the surface area function is a parabola that opens upward. The vertex of the parabola is at ${$ t = -\frac{12}{\frac{3}{2}} = -8$}$, which is not a valid time. This means that the surface area of the ice cube is always increasing over time.

Conclusion

In this article, we explored the concept of surface area in the context of an ice cube. We examined how the side length of the ice cube changes over time, and how this affects its surface area. We also discussed the function ${$ A(s) = 6s^2$}$, which describes the surface area of the ice cube in terms of its side length. By combining the side length function and the surface area function, we were able to determine the surface area of the ice cube at any given time. Finally, we graphed the surface area function to get a better understanding of how the surface area of the ice cube changes over time.

References

[1] "Mathematics for the Nonmathematician" by Morris Kline
[2] "Calculus" by Michael Spivak
[3] "Geometry" by I.M. Gelfand

Appendix

A.1. Derivation of the Surface Area Function

To derive the surface area function, we start with the formula for the surface area of a cube:

${$ A = 6s^2$}$

where ${$ s$}$ is the side length of the cube.

We can substitute the side length function ${$ s(t) = \frac{1}{2} t + 2$}$ into this formula to get:

${$ A(t) = 6\left(\frac{1}{2} t + 2\right)^2$}$

Expanding this expression, we get:

${$ A(t) = 6\left(\frac{1}{4} t^2 + 2t + 4\right)$}$

Simplifying this expression, we get:

${$ A(t) = \frac{3}{2} t^2 + 12t + 24$}$

This is the surface area function that we used in our analysis.

A.2. Graphing the Surface Area Function

$To graph the surface area function, we can use a graphing calculator or a computer algebra system. The graph of the surface area function is a parabola that opens upward. The vertex of the parabola is at ${$t = -\frac{12}{\frac{3}{2}} = -8\$}$, which is not a valid time. This means that the surface area of the ice cube is always increasing over time.<br/> **Q&A: An Ice Cube's Surface Area**$

Q: What is the side length function of the ice cube?

A: The side length function of the ice cube is given by ${$ s(t) = \frac{1}{2} t + 2$}$, where ${$ t$}$ is in hours.

Q: What is the surface area function of the ice cube?

A: The surface area function of the ice cube is given by ${$ A(s) = 6s^2$}$, where ${$ s$}$ is the side length of the ice cube.

Q: How does the side length of the ice cube change over time?

A: The side length of the ice cube increases at a rate of ${$ \frac{1}{2}$}$ inch per hour. In other words, for every hour that passes, the side length of the ice cube increases by ${$ \frac{1}{2}$}$ inch.

Q: How does the surface area of the ice cube change over time?

A: The surface area of the ice cube increases at a much faster rate than its side length. This is because the surface area function is proportional to the square of the side length.

Q: What is the surface area of the ice cube at time ${$ t = 0$}$?

A: At time ${$ t = 0$}$, the side length of the ice cube is ${$ s(0) = \frac{1}{2} (0) + 2 = 2$}$ inches. Therefore, the surface area of the ice cube at time ${$ t = 0$}$ is ${$ A(2) = 6(2)^2 = 24$}$ square inches.

Q: What is the surface area of the ice cube at time ${$ t = 1$}$?

A: At time ${$ t = 1$}$, the side length of the ice cube is ${$ s(1) = \frac{1}{2} (1) + 2 = 2.5$}$ inches. Therefore, the surface area of the ice cube at time ${$ t = 1$}$ is ${$ A(2.5) = 6(2.5)^2 = 56.25$}$ square inches.

Q: How can we determine the surface area of the ice cube at any given time?

A: We can determine the surface area of the ice cube at any given time by substituting the side length function into the surface area function. This gives us the surface area function ${$ A(t) = \frac{3}{2} t^2 + 12t + 24$}$.

Q: What is the graph of the surface area function?

A: The graph of the surface area function is a parabola that opens upward. The vertex of the parabola is at ${$ t = -\frac{12}{\frac{3}{2}} = -8$}$, which is not a valid time. This means that the surface area of the ice cube is always increasing over time.

Q: What are some real-world applications of the surface area function?

A: The surface area function has many real-world applications, such as:

Calculating the surface area of a building or a bridge
Determining the amount of material needed to cover a surface
Estimating the cost of painting or coating a surface
Calculating the surface area of a container or a vessel

Q: What are some limitations of the surface area function?

A: The surface area function has some limitations, such as:

It assumes that the surface is a perfect cube
It does not take into account the shape of the surface
It does not account for any irregularities or imperfections on the surface

Q: How can we modify the surface area function to account for these limitations?

A: We can modify the surface area function to account for these limitations by:

Using a more complex formula that takes into account the shape of the surface
Adding a correction factor to account for any irregularities or imperfections on the surface
Using a different method to calculate the surface area, such as using a computer simulation or a 3D model.