Inverse Variation Problem Solving For N When M Equals 13.5

Hey math enthusiasts! Ever stumbled upon a problem that seems like a puzzle but is actually quite elegant once you understand the core concept? Today, we're diving deep into the world of inverse variation, tackling a classic problem involving the variables m and n. Let's break it down step by step, ensuring you not only grasp the solution but also the underlying principles.

Decoding Inverse Variation

So, what exactly does it mean when we say that "the value of m varies inversely as the square of n"? In mathematical terms, this translates to a beautiful relationship: m is proportional to 1/n². This proportionality is the key to unlocking the solution. Inverse variation basically means that as one variable increases, the other decreases, but in a very specific way. In this case, m doesn't just decrease as n increases; it decreases proportionally to the square of n. This is a crucial detail that we'll use to set up our equation.

To express this relationship mathematically, we introduce a constant of proportionality, often denoted by k. This constant acts as a bridge, linking m and 1/n² with a precise numerical value. Our inverse variation statement then transforms into a powerful equation: m = k/ n². This equation is the cornerstone of our solution. It tells us that m is equal to some constant k divided by the square of n. Understanding this equation is paramount, guys, as it allows us to translate the word problem into a workable mathematical expression. We're not just dealing with abstract concepts here; we're building a concrete formula that will guide us to the answer. Now that we have our equation, we're ready to roll up our sleeves and start plugging in the given values to find the constant of proportionality, k. This is where the real fun begins, as we see how the numbers interact and lead us to the solution. Remember, math isn't just about memorizing formulas; it's about understanding the relationships between variables and using those relationships to solve problems. And that's exactly what we're doing here, unraveling the mystery of inverse variation one step at a time.

Finding the Constant of Proportionality (k)

Now, the problem gives us a crucial piece of information: when n = 3, m = 6. This is our golden ticket! We can use these values to pinpoint the exact value of our constant of proportionality, k. Remember our equation, m = k/ n²? Let's plug in the given values: 6 = k/3². This simple substitution transforms our equation into an algebraic expression we can easily solve. The beauty of this step is that we're moving from a general relationship to a specific numerical connection. We're no longer dealing with abstract variables; we're working with concrete numbers. To isolate k, we multiply both sides of the equation by 3², which is 9. So, we have 6 * 9 = k, which simplifies to k = 54. Bingo! We've found our constant of proportionality. This value, 54, represents the unchanging link between m and n in this particular inverse variation relationship. It's like the secret ingredient in our equation, the key that unlocks the connection between the two variables. Now that we know k, we can rewrite our equation with even more precision: m = 54/n². This equation is now fully personalized to the problem at hand. It's not just a general formula for inverse variation; it's the specific equation that governs the relationship between m and n in this scenario. With this powerful equation in hand, we're ready to tackle the next part of the problem, which asks us to find the value of n when m = 13.5. We've laid the groundwork, and now it's time to put our equation to the test and solve for the unknown. Remember, every step we take builds on the previous one, and our understanding deepens as we progress through the problem.

Solving for n when m = 13.5

Alright, the final part of our adventure! The problem asks us to find the positive value of n when m = 13.5. We've already established our equation: m = 54/n². Now, we simply substitute m = 13.5 into the equation: 13.5 = 54/n². This substitution turns our problem into a straightforward algebraic equation. We're replacing the variable m with its given value, allowing us to isolate and solve for n. To solve for n², we can multiply both sides of the equation by n² and then divide both sides by 13.5. This gives us n² = 54/13.5. Now, let's simplify that fraction. 54 divided by 13.5 equals 4. So, we have n² = 4. We're getting closer to our answer! Remember, we're looking for the value of n, not n². To find n, we need to take the square root of both sides of the equation. The square root of n² is simply n, and the square root of 4 is both 2 and -2. However, the problem specifically asks for the positive value of n. Therefore, our final answer is n = 2. We did it! We've successfully navigated the inverse variation relationship, found the constant of proportionality, and solved for the unknown value of n. This problem showcases the power of mathematical equations to describe and solve real-world relationships. And remember, guys, the key to mastering these concepts is practice and a clear understanding of the underlying principles.

The Answer and Key Takeaways

So, the positive value of n when m = 13.5 is 2 (Option C). This journey through inverse variation highlights a few key takeaways:

• Understanding the definition: Grasp the meaning of inverse variation and how it translates into a mathematical equation.
• Finding the constant of proportionality: Use given information to determine the specific relationship between the variables.
• Substituting and solving: Skillfully manipulate the equation to isolate and solve for the unknown variable.

By mastering these steps, you'll be well-equipped to tackle any inverse variation problem that comes your way. Keep practicing, keep exploring, and keep the math magic alive!

Practice Problems and Further Exploration

To solidify your understanding, try these practice problems:

1. The force of gravity between two objects varies inversely as the square of the distance between them. If the force is 10 Newtons when the distance is 2 meters, what is the force when the distance is 4 meters?
2. The time it takes to complete a task varies inversely with the number of workers. If it takes 6 workers 4 hours to complete a task, how long will it take 8 workers to complete the same task?

Exploring these problems will help you see the versatility of inverse variation in various real-world scenarios. Remember, math is not just about solving equations; it's about understanding the patterns and relationships that govern the world around us. So, keep exploring, keep questioning, and keep building your mathematical skills!

What is the positive value of n when m = 13.5, given that m varies inversely as the square of n and when n = 3, m = 6?