Scientific Notation Explained Expressing Fantasia Sales

Jul 29, 2025 by ADMIN 56 views

Understanding Scientific Notation and Walt Disney's Fantasia Sales

In this article, we will delve into the concept of scientific notation and its application in representing large numbers. We will also address a specific question regarding the sales figures of Walt Disney's Fantasia, as reported by the Wall Street Journal. Understanding scientific notation is crucial in various fields, including science, engineering, and mathematics, as it provides a concise and efficient way to express very large or very small numbers. This article aims to provide a comprehensive explanation of scientific notation, demonstrate its practical use, and guide you through solving the problem presented regarding Fantasia's sales figures. By the end of this article, you will not only be able to convert numbers into scientific notation but also appreciate its significance in real-world applications. We will explore the components of scientific notation, including the coefficient and the exponent, and how they work together to represent a number. Furthermore, we will discuss the advantages of using scientific notation, such as its ability to simplify calculations and make large numbers more manageable. This foundational knowledge will help you tackle similar problems and enhance your understanding of numerical representation.

Scientific Notation Explained

Scientific notation is a standardized way of expressing numbers as the product of a coefficient and a power of 10. The coefficient is a number typically between 1 and 10 (including 1 but excluding 10), and the power of 10 indicates the magnitude of the number. This method is particularly useful for representing extremely large or small numbers concisely. For instance, the number 1,000,000 can be written as 1 x 10^6 in scientific notation. This representation is much more compact and easier to handle than the standard decimal form. The exponent in the power of 10 tells us how many places the decimal point needs to be moved to the left (for positive exponents) or right (for negative exponents) to obtain the original number. Understanding the mechanics of scientific notation is essential for accurately converting numbers between standard and scientific forms. The ability to express numbers in scientific notation is not just a mathematical skill but a practical tool used in various scientific and technical fields. From astronomy, where distances are measured in light-years, to chemistry, where the masses of atoms are incredibly small, scientific notation provides a convenient way to handle these extreme values. Furthermore, scientific notation simplifies calculations involving very large or small numbers, reducing the risk of errors and making the numbers easier to compare and analyze. The process of converting a number to scientific notation involves identifying the coefficient and the appropriate power of 10, which requires a clear understanding of place values and decimal movements. This section aims to provide a solid foundation in the principles of scientific notation, preparing you for the practical application in the following sections.

Applying Scientific Notation to Fantasia Sales

Now, let's apply our understanding of scientific notation to the given problem concerning the sales figures of Walt Disney's Fantasia. According to the Wall Street Journal, Fantasia has sold 14,200,000 copies to date. Our task is to express this number in scientific notation. To do this, we need to identify the coefficient and the power of 10 that will accurately represent this number. First, we place the decimal point after the first non-zero digit, which gives us 1.42. This will be our coefficient. Next, we count the number of places we moved the decimal point to the left to get 1.42 from 14,200,000. We moved the decimal point 7 places, so the power of 10 will be 10^7. Therefore, 14,200,000 in scientific notation is 1.42 x 10^7. This process demonstrates the practical application of scientific notation in representing large numbers. By converting the sales figures to scientific notation, we can easily grasp the magnitude of the number and compare it with other large values. Scientific notation makes it easier to work with such numbers in calculations and comparisons. In this case, expressing Fantasia's sales in scientific notation provides a clear and concise representation of its commercial success. Understanding how to convert real-world numbers like sales figures into scientific notation is a valuable skill, applicable in various contexts, from business and finance to science and engineering. This example highlights the importance of mastering scientific notation for effective communication and analysis of numerical data.

Analyzing the Given Options

To solve the problem, we need to analyze the options provided and determine which one correctly expresses 14,200,000 in scientific notation. The options are:

A. 1.42 x 10^6 B. 14.2 x 10^6 C. 142 x 10^5 D. 1.42 x 10^7

Let's examine each option:

Option A: 1.42 x 10^6 – This represents 1.42 multiplied by 1,000,000, which equals 1,420,000. This is significantly less than 14,200,000, so this option is incorrect.
Option B: 14.2 x 10^6 – While the digits are correct, the coefficient 14.2 is not between 1 and 10. Therefore, this is not a correct representation in scientific notation, although it does equal 14,200,000.
Option C: 142 x 10^5 – Similarly, the coefficient 142 is not between 1 and 10, making this an incorrect representation in scientific notation. This option also equals 14,200,000, but it doesn't adhere to the scientific notation format.
Option D: 1.42 x 10^7 – This represents 1.42 multiplied by 10,000,000, which equals 14,200,000. The coefficient 1.42 is between 1 and 10, and the power of 10 is correct. Therefore, this is the correct representation in scientific notation.

By systematically analyzing each option, we can identify the correct answer. This process reinforces the importance of understanding the rules of scientific notation and applying them carefully. The correct option, D, demonstrates the accurate conversion of a large number into scientific notation, highlighting the practical application of this mathematical concept.

The Correct Answer: D. 1.42 x 10^7

Based on our analysis, the correct answer is D. 1.42 x 10^7. This option accurately represents 14,200,000 in scientific notation. The coefficient, 1.42, is a number between 1 and 10, and the exponent, 7, indicates that the decimal point has been moved seven places to the left. This representation not only adheres to the rules of scientific notation but also provides a clear and concise way to express the large number of Fantasia copies sold. Choosing the correct answer highlights the importance of understanding the fundamental principles of scientific notation, including the proper format and the role of the coefficient and exponent. This skill is crucial for accurately representing and manipulating large and small numbers in various scientific and mathematical contexts. By correctly identifying the sales figure in scientific notation, we demonstrate the practical application of this concept in real-world scenarios, such as reporting sales data. The process of arriving at the correct answer reinforces the need for careful analysis and attention to detail when working with numerical representations. This understanding of scientific notation enables us to communicate numerical information effectively and efficiently, making it an invaluable tool in both academic and professional settings.

Conclusion

In conclusion, this article has provided a comprehensive explanation of scientific notation and its application in representing large numbers, specifically the sales figures of Walt Disney's Fantasia. We have explored the principles of scientific notation, including the importance of the coefficient and the exponent, and how they work together to express a number in a concise and standardized format. By converting the sales figure of 14,200,000 copies into scientific notation, we demonstrated the practical utility of this concept in real-world scenarios. The correct representation, 1.42 x 10^7, not only adheres to the rules of scientific notation but also provides a clear and easily understandable way to express this large number. Understanding and applying scientific notation is a valuable skill in various fields, including mathematics, science, and engineering. It allows for the efficient handling of very large and very small numbers, simplifying calculations and making data analysis more manageable. This article has aimed to provide a solid foundation in scientific notation, equipping you with the knowledge and skills to tackle similar problems and appreciate its significance in numerical representation. Mastering scientific notation enhances your ability to communicate numerical information effectively and accurately, which is crucial in both academic and professional settings. The example of Fantasia's sales highlights the relevance of this concept in everyday contexts, reinforcing its importance in our understanding of the world around us.