Question

True or False? In Exercises \(67-70\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. In linear growth, the rate of growth is constant

Short answer

The statement 'In linear growth, the rate of growth is constant' is True.

Step-by-step solution

Understanding Linear Growth

In mathematics, linear growth refers to a growth that occurs at a constant rate over time. It creates a straight line when plotted on a graph.

Evaluating the Statement

Next, we compare this understanding of linear growth with the statement given in the exercise. The statement claims that in linear growth, the rate of growth is constant. This aligns with the definition of linear growth, as in such a pattern of growth, the quantity increases by the same amount over each equal interval of time.

Finalizing the Result

Since the statement aligns with the definition of linear growth, it can be concluded that the statement 'In linear growth, the rate of growth is constant' is true.

Key concepts

Constant Rate of Growth

Understanding the constant rate of growth is fundamental in grasping different types of progression in mathematics. It means that a value increases by a fixed amount in equal intervals of time. This is central to many real-world phenomena, such as the interest you earn in a savings account, or the distance covered by a car traveling at a steady speed.

Imagine adding the same number of bricks each day to a growing stack, ultimately resulting in a tower. The height of the stack increases by the exact number of bricks you add — this is a perfect example of a constant rate of growth. Analyzing and predicting behaviors of such growth can be essential not just in mathematics, but in finance, computer science, and various sciences where a predictable increase is observed.

In our textbook exercise, the concept is applied to the realm of linear growth. When plotted on a graph, this constant increase presents itself as a straight line, hence the term 'linear'.

Mathematics

Mathematics is the language of patterns and relationships among quantities, shapes, and structures. It's not just about numbers; it's about understanding and describing the world in a systematic way. Within this broad field, growth patterns are a particular area of interest. These patterns help mathematicians and scientists describe how things change over time.

To give additional context, consider baking a cake. The recipe you follow is a kind of algorithm, a systematic mathematical set of instructions that, when followed, yields consistent results. Just as algorithms and recipes bring clarity and predictability, the mathematical concept of constant rate of growth brings a clear, easy-to-follow pattern to many aspects of life and science.

The exercise from the textbook involved a true or false question relating to linear growth — a concept that is a small, yet significant part of the vast and intricate world of mathematics.

Linear Functions

A linear function is one of the most basic and important concepts in algebra and economics. The function is represented by the equation of a straight line, typically written as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In simple terms, the slope \( m \) represents the rate of change, and the y-intercept \( b \) represents the starting value when \( x \) is zero.

In the context of our exercise, where linear growth means a constant rate of growth, the slope \( m \) remains consistent. This consistency affirms that each unit increase in \( x \), the independent variable, results in a predictable and constant increase in \( y \), the dependent variable. Recognizing this allows students to anticipate outcomes in linear scenarios, such as budgeting monthly expenses or understanding the path of a moving object.

As you delve into plotting these functions or solving for variables within them, remember that linear equations are the foundation for much more complex mathematical functions and models, all of which serve to interpret our ever-changing world.