Understanding The Function H(t)=210-15t

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In the realm of mathematics, functions serve as fundamental building blocks for modeling relationships between variables. Understanding the components of a function and how they interact is crucial for solving problems and making predictions in various fields, from physics and engineering to economics and computer science. In this article, we will delve into the function h(t) = 210 - 15t, dissecting its components and exploring its behavior. This analysis will provide a solid foundation for comprehending the nature of linear functions and their applications.

Dissecting the Function: Identifying Key Components

To effectively analyze the function h(t) = 210 - 15t, it's essential to identify its core components. Let's break it down:

  • Function Name: The function is denoted by the letter h. This name serves as a label, allowing us to refer to the function concisely. It's like giving a name to a specific process or relationship we want to study.
  • Input Variable: The input variable is represented by t. This variable is the independent element that we feed into the function. Its value determines the output of the function. Think of it as the raw material that the function processes.
  • Output Variable: The output variable is represented by h(t). This is the dependent variable, as its value depends on the input value t. h(t) represents the result of applying the function h to the input t. It's the finished product that the function generates.
  • Constants: The function includes two constants: 210 and -15. Constants are fixed values that don't change with the input variable. In this case, 210 represents the initial value or starting point, while -15 represents the rate of change or slope.

The Role of Variables: Input vs. Output

The distinction between input and output variables is paramount in understanding functions. The input variable, often referred to as the independent variable, is the value we control or choose. It's the variable we manipulate to observe its effect on the output. In contrast, the output variable, also known as the dependent variable, is the result of applying the function to the input. Its value is determined by the input value and the function's rule.

In the context of h(t) = 210 - 15t, t is the input variable, and h(t) is the output variable. This means that we can choose different values for t and observe how the function transforms them into corresponding values for h(t). For example, if we input t = 0, the output is h(0) = 210 - 15(0) = 210. If we input t = 5, the output is h(5) = 210 - 15(5) = 135. This demonstrates how the input value influences the output value through the function's rule.

Unveiling the Nature of the Function: Linear Functions

Now that we've dissected the components of h(t) = 210 - 15t, let's delve into the nature of the function itself. This function is a classic example of a linear function. Linear functions are characterized by a constant rate of change, meaning that the output changes by a fixed amount for every unit change in the input. This constant rate of change is represented by the slope of the function's graph, which is a straight line.

Identifying Linear Functions: Key Characteristics

Linear functions possess several key characteristics that distinguish them from other types of functions:

  • Constant Rate of Change: As mentioned earlier, linear functions exhibit a constant rate of change. This means that the ratio of the change in the output to the change in the input is always the same. This constant rate of change is the slope of the line.
  • Straight-Line Graph: The graph of a linear function is always a straight line. This is a direct consequence of the constant rate of change. A straight line represents a consistent and predictable relationship between the input and output variables.
  • Slope-Intercept Form: Linear functions can be expressed in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept (the point where the line crosses the y-axis). This form provides a clear representation of the function's key parameters.

h(t) = 210 - 15t in Slope-Intercept Form

Our function, h(t) = 210 - 15t, can be easily rearranged into slope-intercept form. By swapping the terms, we get h(t) = -15t + 210. Now, it's clear that the slope m is -15 and the y-intercept b is 210. This means that the line has a negative slope, indicating a decreasing relationship between t and h(t), and it intersects the h(t)-axis at the point (0, 210).

Interpreting the Function: Real-World Applications

Functions are not just abstract mathematical concepts; they have practical applications in modeling real-world phenomena. The function h(t) = 210 - 15t can represent various scenarios, depending on the context. Let's explore a few examples:

Scenario 1: Depreciation of an Asset

Imagine a piece of equipment initially valued at $210 that depreciates at a rate of $15 per year. In this case, h(t) could represent the value of the equipment after t years. The initial value is 210, and the value decreases by 15 for each year that passes. The negative slope of -15 indicates the depreciation rate.

Scenario 2: Water Tank Drainage

Consider a water tank that initially contains 210 gallons of water. If the tank is drained at a rate of 15 gallons per minute, h(t) could represent the amount of water remaining in the tank after t minutes. The initial amount of water is 210, and the amount decreases by 15 gallons per minute. The negative slope again signifies a decreasing quantity.

Scenario 3: Linear Motion

Suppose an object starts at a position of 210 units and moves backward at a rate of 15 units per second. Then h(t) could represent the object's position after t seconds. The initial position is 210, and the object moves 15 units backward each second. The negative slope indicates movement in the negative direction.

Making Predictions with the Function

One of the key benefits of using functions to model real-world situations is the ability to make predictions. Once we have a function like h(t) = 210 - 15t, we can use it to estimate the output for any given input value. For example, in the depreciation scenario, we can predict the value of the equipment after 10 years by substituting t = 10 into the function: h(10) = 210 - 15(10) = 60. This suggests that the equipment will be worth $60 after 10 years.

Conclusion: The Power of Functions in Mathematical Modeling

In this comprehensive guide, we've dissected the function h(t) = 210 - 15t, exploring its components, nature, and applications. We've seen how to identify the function name, input variable, output variable, and constants. We've also learned that this function is a linear function, characterized by a constant rate of change and a straight-line graph. Furthermore, we've explored how this function can model various real-world scenarios, such as depreciation, water tank drainage, and linear motion. By understanding the function's behavior, we can make predictions and gain insights into the relationships between variables.

Functions are powerful tools in mathematics, allowing us to represent and analyze relationships between quantities. By mastering the concepts presented in this article, you'll be well-equipped to tackle more complex mathematical problems and apply your knowledge to real-world situations. The journey into the world of functions is just beginning, and the possibilities are endless. Embrace the challenge, explore the intricacies, and unlock the power of mathematical modeling.