Problem 89 - Master Resource Book in JEE Main Physics

Question

Given \(\pi=3.14\). The value of \(\pi^{2}\) with due regard for significant figures is (a) \(9.86\) (b) \(9.859\) (c) \(9.8596\) (d) \(9.85960\)

Step-by-Step Solution

Answer

The answer is (a) 9.86.

Step 1: Identify the Given Value and its Significant Figures

We are given that \( \pi = 3.14 \). This value has three significant figures: '3', '1', and '4'.

Step 2: Calculate \( \pi^2 \)

To find \( \pi^2 \), we multiply \( \pi \) by itself: \( 3.14 \times 3.14 \). This equals \( 9.8596 \).

Step 3: Determine the Number of Significant Figures in \( \pi^2 \)

Since \( \pi \) was given to three significant figures (3.14), the result \( \pi^2 \) should also be reported to three significant figures.

Step 4: Round \( \pi^2 \) to the Correct Number of Significant Figures

Take the \( \pi^2 \) value calculated (9.8596) and round it to three significant figures. We round 9.8596 to 9.86 to keep three significant figures.

Key Concepts

Calculation Accuracy

When working with numbers in mathematics, calculation accuracy is critical to preserving the integrity of your results. It refers to the degree to which calculated results represent the true value. Maintaining accuracy ensures that the numerical data remains reliable and precise.

In mathematics exercises like calculating \( \pi^2 \), it's essential to execute operations with precision. For instance, multiplying \( 3.14 \) by itself to compute \( \pi^2\) results in \( 9.8596 \). During the process, every step needs accuracy to avoid compound errors that affect the final result.

Some ways to enhance calculation accuracy include:

• Meticulously checking each step during the computation.
• Using mathematical tools like calculators for intricate calculations.
• Maintaining attention to detail, especially when dealing with decimals or fractional values.

Rounding Numbers

Rounding numbers is a fundamental concept tied closely to the precision required in significant figures. It involves adjusting a number to reduce its digits while maintaining its value to the nearest practical level.

When \( 9.8596 \) is rounded to three significant figures, it becomes \( 9.86 \). This process involves looking at the fourth digit (in this case, 5), which is 5 or more, prompting us to round up the last significant digit.

Here are key points for rounding numbers:

• Locate the digit at the desired significant figure.
• Observe the digit directly following this; if it's 5 or greater, round up.
• Keep in mind that rounding helps in simplifying numbers while retaining reasonable accuracy.

Mathematics Education

Learning about significant figures and rounding in mathematics education is crucial for students. These concepts help bridge the gap between pure theoretical math and real-world applications. Understanding how to handle such numbers empowers students to work with scientific data and everyday scenarios involving measurements.

In classrooms, educators often introduce exercises to enhance problem-solving skills using significant figures. This practical approach supports students in grasping the importance of accuracy and precision.

Educational methods to teach these concepts include:

• Real-life examples, such as dealing with finances, where precision impacts outcomes.
• Step-by-step problems like calculating \( \pi^2 \), where students practice rounding and accuracy.
• Interactive activities that simulate measurement scenarios requiring precise calculations.