Problem 37 - Precalculus

Question

Say whether the statement is TRUE or FALSE. (In Exercises \(37-40\), do not use a calculator or table; use instead the approximations \(\sqrt{2} \approx 1.4 \text { and } \pi \approx 3.1 .)\) $$2 \pi<6$$

Step-by-Step Solution

Answer

The statement is FALSE.

Step 1: Substitute the Approximation for \( \pi \)

We know that \( \pi \approx 3.1 \). Substitute \( 3.1 \) for \( \pi \) to approximate \( 2 \pi \), giving us \( 2 \times 3.1 \).

Step 2: Perform the Multiplication

Calculate the product of \( 2 \) and the approximation \( 3.1 \). Thus, we have \( 2 \times 3.1 = 6.2 \).

Step 3: Compare the Result with 6

The result of our multiplication is \( 6.2 \). Compare this result to \( 6 \). Since \( 6.2 > 6 \), this implies that \( 2 \pi \) is greater than \( 6 \).

Step 4: Conclusion Based on Comparison

Since \( 2 \pi > 6 \), the statement \( 2 \pi < 6 \) is FALSE.

Key Concepts

Approximations

The concept of approximations is fundamental in mathematics, as it allows us to simplify complex numbers into more manageable forms. In the original exercise, we use approximations to evaluate expressions without a calculator or table. Approximations work by providing us with a value that is close enough to the actual number, allowing us to perform calculations easily.

• For example, \( \pi \) is approximated to \( 3.1 \) and \( \sqrt{2} \) to \( 1.4 \).
• These simplified numbers are particularly useful when we want a quick reference or when performing mental arithmetic.

Remember, approximations are not exact. However, they give us a good estimate, allowing us to assess and answer questions efficiently, especially in exams or timed quizzes. By approximating, we can visualize and analyze situations better, like comparing \( 2\pi \) with 6, as we did in the solution.

Mathematical Reasoning

Mathematical reasoning involves logically analyzing and connecting information to reach conclusions. In mathematical problems, reasoning helps us establish relationships between numbers and mathematical concepts. For the given exercise, systematic reasoning is crucial.

• First, we recognize that we need to replace \( \pi \) with its approximation, \( 3.1 \), to facilitate our comparison.
• Using reasoning, we multiply \( 2 \) by \( 3.1 \) to approximate \( 2\pi \), leading us to a computed value of \( 6.2 \).

This kind of reasoning is logical, marking out each step clearly and ensuring that we back our steps with proper mathematical operations. It helps in understanding how and why our approach leads to concluding \( 2\pi > 6 \), thus deeming the statement \( 2\pi < 6 \) false.

Comparison of Numerical Values

Comparing numerical values is a key skill in mathematics that is often used to determine the relative size of numbers. In this context, we need to compare our calculated approximation of \( 2\pi \) with 6.

• After calculating \( 2 \times 3.1 = 6.2 \), we need to decide whether \( 6.2 \) is less than or greater than \( 6 \).
• By simple comparison, \( 6.2 \) is clearly greater than \( 6 \), indicating the inequality holds as true for the opposite statement, \( 2\pi > 6 \).

Effective comparison does not only involve finding if one number is greater or less. It requires understanding why this relationship matters in the context of a mathematical statement, helping refine accuracy in problem-solving and verifying assumptions made through approximations and reasoning. Thus, confirming the falsity of the original statement through the examination of our approximated values.