Problem 40 - 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 \(\sqrt{2} \approx 1.4 \text { and } \pi \approx 3.1 .)\) $$\pi^{2}<12$$

Step-by-Step Solution

Answer

The statement is TRUE.

Step 1: Understanding the Statement

We need to determine if the expression \( \pi^2 < 12 \) is true or false using the approximation \( \pi \approx 3.1 \).

Step 2: Evaluate \( \pi^2 \) Using the Approximation

First, we substitute the approximation \( \pi \approx 3.1 \) into \( \pi^2 \). This gives us \( (3.1)^2 \).

Step 3: Calculate \( (3.1)^2 \)

To calculate \( (3.1)^2 \), multiply 3.1 by itself: \( 3.1 \times 3.1 = 9.61 \).

Step 4: Compare \( \pi^2 \) with 12

We now compare the result \( 9.61 \) with 12. Since 9.61 is less than 12, \( \pi^2 < 12 \).

Step 5: State the Conclusion

Based on the calculation, the statement \( \pi^2 < 12 \) is TRUE.

Key Concepts

Approximations

Approximations are a useful mathematical tool that help simplify complex problems. When exact values are not required, or when they aren't readily accessible, using approximations can make calculations more manageable. In the given exercise, we used an approximation for \( \pi \) to evaluate a statement.

Instead of using the exact value of \( \pi \), which is an irrational number approximately equal to 3.14159, we apply the approximation \( \pi \approx 3.1 \). This simplified value allows for quick mental calculations and a rough estimate that can be useful for verifying logical statements. Approximations are particularly practical in:

• Testing mathematical ideas without the need for a calculator.
• Making quick decisions when exact precision isn't necessary.
• Facilitating problem-solving by transforming irrational numbers into more manageable figures.

The key takeaway is that approximations are all about using a close-enough value to simplify the process, while still getting a good sense of the outcome.

Exponents

Exponents are a mathematical notational shorthand used to express repeated multiplication of a number by itself. When dealing with exponents, it's vital to understand how they work and their impact on numbers. An exponent is written as a superscript to the right of a base number.

For example, in the expression \( x^2 \), \( x \) is the base, and \( 2 \) is the exponent. This means \( x \) is multiplied by itself: \( x \times x \). In the exercise provided, \( \pi^2 \) represents \( \pi \) multiplied by itself. Using the approximation \( \pi \approx 3.1 \), we calculate \( 3.1^2 \), which equals 9.61.

Some handy tips for working with exponents are:

• Exponentiation is commutative for addition but not for multiplication. For example, \( (a+b)^2 eq a^2 + b^2 \).
• Keep track of signs. Negative bases with even exponents yield positive results.
• Any number to the power of zero is \( 1 \), except when the base is zero.

Understanding exponents is crucial in simplifying and solving many mathematical expressions, also helping with intuitive grasp of how numbers scale with repeated multiplication.

Mathematical Reasoning

Mathematical reasoning involves logical thinking to solve problems and prove or disprove statements. It helps in making sense of mathematical tasks by using both numerical evidence and logical analysis.

In this exercise, we employed mathematical reasoning to evaluate whether the statement \( \pi^2 < 12 \) is true. We started by understanding what the expression means and then used an approximation for \( \pi \). By calculating \( 3.1^2 \) and comparing it to 12, we logically concluded that 9.61 is indeed less than 12.

Some key elements of mathematical reasoning include:

• Identifying and understanding the problem or statement.
• Breaking the problem into manageable parts.
• Using logical steps and operations to analyze the problem.
• Drawing conclusions based on gathered evidence and calculations.

Mathematical reasoning is a critical skill that allows us to draw valid conclusions about mathematical concepts, enhancing problem-solving abilities and supporting confident decision-making in mathematics.