Understanding A 747 Airplane's Weight Using Scientific Notation

A 747 airplane is a marvel of engineering, a massive machine that can carry hundreds of passengers and tons of cargo across vast distances. But just how heavy is one of these behemoths? The question tells us that a 747 airplane weighs about 600,000 pounds. This is an impressive figure, but to truly grasp the scale of this weight, we can express it in different units and use scientific notation, a powerful tool for representing very large or very small numbers. The question further states that the weight can also be expressed as approximately $3 imes 10^n$ tons, where 'n' is an exponent we need to determine. Our task is to find the most reasonable value of 'n'. This exploration will not only help us understand the weight of a 747 but also solidify our understanding of unit conversions and scientific notation, essential concepts in mathematics and science. The heart of this problem lies in converting pounds to tons and then expressing the result in scientific notation. We'll need to use the given conversion factor (2,000 pounds per ton) and manipulate the numbers to fit the form $3 imes 10^n$. This involves understanding how the exponent 'n' affects the magnitude of the number and how to round numbers appropriately when dealing with approximations. This problem is more than just a calculation; it's an exercise in estimation and understanding the relative sizes of different units. By working through it, we'll gain a deeper appreciation for the scale of everyday objects and the power of mathematical tools to represent them effectively. We will also hone our skills in problem-solving, a crucial ability in various fields, from engineering and physics to finance and economics.

The Conversion Process Pounds to Tons

The first step in solving this problem is converting the weight from pounds to tons. We're given that 1 ton is equal to 2,000 pounds. To convert 600,000 pounds to tons, we need to divide 600,000 by 2,000. This calculation will tell us how many tons the 747 weighs. So, we perform the division: 600,000 pounds / 2,000 pounds/ton = 300 tons. This means the 747 weighs approximately 300 tons. Now that we have the weight in tons, we can move on to expressing it in scientific notation. Scientific notation is a way of writing numbers as a product of a number between 1 and 10 and a power of 10. This is particularly useful for very large or very small numbers, as it makes them easier to read and compare. In our case, we need to express 300 tons in the form $3 imes 10^n$. To do this, we need to identify the appropriate power of 10. We know that 300 can be written as 3 multiplied by 100. And 100 is simply $10^2$. Therefore, 300 tons can be written as $3 imes 10^2$ tons in scientific notation. This step is crucial because it directly relates the weight in tons to the desired form $3 imes 10^n$. By understanding the relationship between 300 and $3 imes 10^2$, we can easily identify the value of 'n'. The conversion process from pounds to tons is a fundamental skill in many applications, from everyday life to scientific calculations. It's important to understand the units involved and how to convert between them accurately. Scientific notation, on the other hand, is a powerful tool for representing numbers in a concise and manageable way, especially when dealing with very large or very small quantities. Mastering these concepts is essential for anyone working with numbers in any field.

Deciphering Scientific Notation Finding the Value of 'n'

Now that we've converted the weight of the 747 to 300 tons, the next critical step is to express this value in scientific notation. Scientific notation, as we've discussed, is a standardized way of writing numbers as a product of a number between 1 and 10 and a power of 10. This form is incredibly useful for representing very large or very small numbers, making them easier to handle and compare. The question provides us with the expression $3 imes 10^n$ tons, and our goal is to determine the value of 'n'. This 'n' represents the exponent, the power to which 10 is raised. The exponent essentially tells us how many places to move the decimal point to the right or left to get the original number. In our case, we have 300 tons. To express this in scientific notation, we need to rewrite it as 3 multiplied by a power of 10. We can rewrite 300 as $3 imes 100$. Now, we need to express 100 as a power of 10. We know that $10^2$ (10 squared) is equal to 100. Therefore, 300 can be written as $3 imes 10^2$. Comparing this with the given expression $3 imes 10^n$, we can see that 'n' must be equal to 2. This means that the most reasonable value of 'n' is 2. Understanding scientific notation is crucial in many scientific and engineering disciplines. It allows us to work with numbers of vastly different magnitudes without getting bogged down by long strings of digits. The exponent in scientific notation provides a quick and easy way to compare the relative sizes of numbers. For example, $1 imes 10^6$ is a million, while $1 imes 10^9$ is a billion. The difference in the exponents (6 vs. 9) tells us that a billion is a thousand times larger than a million. In our problem, finding the value of 'n' helps us understand the scale of the 747's weight in a more manageable way. $3 imes 10^2$ tons is a much easier number to grasp than 600,000 pounds.

Choosing the Correct Answer Among the Options

Having determined the value of 'n' to be 2, we now turn our attention to the multiple-choice options provided in the question. The options are:

A. -2 B. -1 C. 2 D. 4

We've already established that the weight of the 747 in tons, expressed in scientific notation, is approximately $3 imes 10^2$ tons. This directly corresponds to option C, which gives the value of 'n' as 2. The other options can be easily ruled out. Options A and B have negative values for 'n', which would represent very small numbers (fractions of a ton), not the massive weight of an airplane. Option D, with 'n' equal to 4, would represent 30,000 tons, which is significantly heavier than the actual weight of a 747. Therefore, the only logical and correct answer is C. 2. This step is a crucial part of the problem-solving process. It's not enough to just arrive at a numerical answer; we also need to carefully compare our answer with the given options and ensure that it makes sense in the context of the problem. In this case, understanding the magnitude of different powers of 10 is essential for eliminating the incorrect options. Option A (-2) would represent 0.03 tons, which is far too light for an airplane. Option B (-1) would be 0.3 tons, still too light. Option D (4) would be 30,000 tons, which is an implausibly high weight for a 747. By systematically analyzing the options and comparing them with our calculated value, we can confidently select the correct answer. This process reinforces the importance of estimation and number sense in problem-solving. It's always a good idea to ask yourself if your answer seems reasonable in the real world. In this case, a weight of 300 tons for a 747 aligns with our general understanding of the size and weight of these aircraft.

Conclusion The Value of 'n' and the Weight of a 747

In conclusion, by converting the weight of a 747 airplane from pounds to tons and then expressing it in scientific notation, we have successfully determined the most reasonable value of 'n' in the expression $3 imes 10^n$ tons. The answer is C. 2. This means that the weight of the 747 can be approximated as $3 imes 10^2$ tons, or 300 tons. This exercise has demonstrated the power of scientific notation in representing large numbers in a concise and manageable form. It has also highlighted the importance of unit conversions in solving real-world problems. By understanding these concepts, we can better grasp the scale of objects and phenomena around us, from the weight of an airplane to the distances between stars. The problem-solving process we've used here can be applied to a wide range of quantitative challenges. Breaking down a problem into smaller steps, identifying the relevant information, and applying appropriate mathematical tools are all essential skills for success in mathematics, science, and beyond. Furthermore, this problem has reinforced the importance of critical thinking and estimation. By considering the magnitude of the numbers involved and comparing our answer with the given options, we were able to confidently select the correct solution. This ability to reason about numbers and make informed judgments is a valuable asset in all aspects of life. So, the next time you see a 747 soaring through the sky, remember that it weighs approximately 300 tons, or $3 imes 10^2$ tons, a testament to human engineering and the power of mathematics to describe the world around us. Understanding the weight of a 747 in this context, using scientific notation, provides a tangible example of how mathematics can help us understand and appreciate the scale of the world around us. It's a reminder that even seemingly abstract concepts like scientific notation have practical applications in everyday life.