Problem 134 - College Algebra

Question

In Exercises 132–135, determine whether each statement makes sense or does not make sense, and explain your reasoning. The population of Colorado is approximately \(4.6 \times 10^{12}\)

Step-by-Step Solution

Answer

The statement does not make sense. The total world population is just over 7 billion, so a population of \(4.6\times 10^{12}\) for a single state is not plausible.

Step 1: Understand the Context

This exercise is asking for reasoning whether or not a population of \(4.6 \times 10^{12}\) for the state of Colorado makes sense. The first step is to understand what this number represents. In scientific notation, \(4.6 \times 10^{12}\) represents 4,600,000,000,000.

Step 2: Analyze The Given Number

The number \(4.6 \times 10^{12}\) implies there are four trillion six hundred billion people in Colorado. It is pertinent at this stage to understand the approximate human population of the world. As of current estimates, the entire world's human population is somewhat over 7 billion. This indicates that the given population for Colorado, a single state of a single country, far exceeds the total global population, thus evidencing its implausibility.

Step 3: Draw a Conclusion

After comparing the proposed population for Colorado to the global human population, it's easy to conclude that the number \(4.6 \times 10^{12}\) is extraordinarily high, and therefore, does not make sense.

Key Concepts

Reasoning with Scientific Notation

Scientific notation is a powerful tool in algebra that allows for the expression of very large or very small numbers in a compact form. It is written as the product of a number between 1 and 10 and a power of 10. For instance, when we see a number like \(4.6 \times 10^{12}\), it's expressing a very large quantity; in this case, 4.6 trillion.

When reasoning with scientific notation, it is critical to understand the scale of the numbers you are dealing with. Comparing these numbers to real-world quantities is essential. In the provided exercise, understanding that the world's population is around \(7.8 \times 10^9\) clarifies that a state population of \(4.6 \times 10^{12}\) is not reasonable or sensible. Reasoning with scientific notation requires not just arithmetic skills but also a broader awareness of the context the numbers are used within.

Population Estimation

Population estimation is a common application of algebra and scientific notation in understanding demographics. To estimate populations, especially for large areas or groups, scientific notation offers a clear and concise way to represent numbers.

In our example, comparing the given figure of \(4.6 \times 10^{12}\) for Colorado's population to real population data enables us to estimate the accuracy of such a number. Knowing the approximate human population of Earth can put into perspective whether a given estimate for a smaller subset (like a state) is within a plausible range. Therefore, understanding the order of magnitude, which is basically the power of ten in scientific notation, plays a critical role in correctly estimating and evaluating population figures.

Analyzing Algebraic Statements

Algebraic statements often include numbers, variables, and operations that represent real-world quantities and relationships. Analyzing these statements involves understanding and interpreting the mathematical language used.

To analyze the statement concerning Colorado's population, one must unpack the scientific notation to appreciate the magnitude of the number. By converting \(4.6 \times 10^{12}\) into a standard numeral or comparing it with other known quantities, it becomes apparent if the statement holds true or not. This is a fundamental skill in algebra: not just being able to manipulate numbers and expressions but also to critically evaluate the statements they represent for logical and factual consistency.

Contextualizing Mathematical Problems

Contextualizing mathematical problems is the practice of situating a numerical or algebraic problem within its broader real-world scenario.

In doing so, one applies their knowledge of the world and the relevant field (like geography, demography, or economics) to interpret the mathematical statements accurately. By understanding that the state of Colorado is part of the United States and that the entire population of the US is a fraction of a billion, it highlights the falsehood of the exercise statement. This exercise showcases the importance of integrating mathematical skills with factual knowledge to make rational judgments and solve problems effectively.