Breaking Down the Function

Oct 13, 2025 by miodeya.com 46 views

Understanding the Function: Unraveling the Mystery of the Number 60

In the world of mathematics, functions are used to describe the relationship between two variables. The function $y=\frac{60}{x}$ is a classic example of a rational function, where $y$ represents the time in hours and $x$ represents the average speed in miles per hour. In this article, we will delve into the world of functions and explore the significance of the number 60 in the given function.

What Does the Number 60 Represent?

The number 60 in the function $y=\frac{60}{x}$ is a crucial component that holds the key to understanding the relationship between time and speed. To unravel the mystery of the number 60, let's break down the function and analyze its components.

Breaking Down the Function

The function $y=\frac{60}{x}$ can be broken down into two main components:

The numerator (60): This is the number that is being divided by the variable $x$ .
The denominator ( $x$ ): This is the variable that is being used to divide the numerator.

The Significance of the Number 60

So, what does the number 60 represent in the function? To answer this question, let's consider the following scenario:

Suppose Michael travels at an average speed of 1 mile per hour. In this case, the value of $x$ would be 1. To find the time it takes for Michael to travel, we can plug in the value of $x$ into the function:

y=\frac{60}{1}=60

As we can see, the value of $y$ is 60, which represents the time in hours. This means that if Michael travels at an average speed of 1 mile per hour, it will take him 60 hours to cover a certain distance.

The Relationship Between Time and Speed

Now that we have a better understanding of the number 60, let's explore the relationship between time and speed. The function $y=\frac{60}{x}$ shows that as the average speed ( $x$ ) increases, the time ( $y$ ) decreases. This is because the function is a rational function, where the numerator is a constant (60) and the denominator is a variable ( $x$ ).

The Inverse Relationship

The function $y=\frac{60}{x}$ exhibits an inverse relationship between time and speed. This means that as the speed increases, the time decreases, and vice versa. This is a fundamental concept in physics and is used to describe the relationship between distance, speed, and time.

In conclusion, the number 60 in the function $y=\frac{60}{x}$ represents the time in hours that it takes for Michael to travel at a certain average speed. The function exhibits an inverse relationship between time and speed, where as the speed increases, the time decreases, and vice versa. This is a fundamental concept in physics and is used to describe the relationship between distance, speed, and time.

Here are some example problems to help you understand the concept better:

Problem 1: If Michael travels at an average speed of 2 miles per hour, how long will it take him to cover a certain distance?
Problem 2: If Michael travels at an average speed of 3 miles per hour, how long will it take him to cover a certain distance?
Problem 3: If Michael travels at an average speed of 4 miles per hour, how long will it take him to cover a certain distance?

Here are the solutions to the example problems:

Problem 1: If Michael travels at an average speed of 2 miles per hour, the value of $x$ would be 2. Plugging this value into the function, we get:

y=\frac{60}{2}=30

This means that it will take Michael 30 hours to cover a certain distance.

Problem 2: If Michael travels at an average speed of 3 miles per hour, the value of $x$ would be 3. Plugging this value into the function, we get:

y=\frac{60}{3}=20

This means that it will take Michael 20 hours to cover a certain distance.

Problem 3: If Michael travels at an average speed of 4 miles per hour, the value of $x$ would be 4. Plugging this value into the function, we get:

y=\frac{60}{4}=15

This means that it will take Michael 15 hours to cover a certain distance.

The function $y=\frac{60}{x}$ has numerous real-world applications in fields such as physics, engineering, and transportation. Here are a few examples:

Physics: The function is used to describe the relationship between distance, speed, and time in physics.
Engineering: The function is used to design and optimize transportation systems, such as roads and highways.
Transportation: The function is used to calculate the time it takes to travel between two points, given the average speed and distance.