Solving Inverse Variation Problems A Step By Step Guide

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In the realm of mathematics, inverse variation is a fundamental concept that describes a relationship between two variables where one variable decreases as the other increases, and vice versa. This relationship is often encountered in various scientific and engineering applications, making it crucial to grasp its underlying principles. When we say that a variable m varies inversely as the square of another variable n, we are essentially stating that their product is a constant. Mathematically, this can be expressed as:

m n² = k,

where k represents the constant of variation. This constant, k, plays a pivotal role in defining the specific inverse relationship between m and n. The inverse square relationship implies that as n increases, m decreases proportionally to the square of n, and conversely, as n decreases, m increases proportionally to the square of n. This relationship can be visualized as a curve that slopes downwards, indicating the inverse nature of the connection between the two variables.

To truly understand inverse variation, it is helpful to contrast it with direct variation. In direct variation, as one variable increases, the other variable also increases proportionally, and vice versa. This is in stark contrast to inverse variation, where the variables move in opposite directions. Examples of inverse variation abound in the real world, such as the relationship between the pressure and volume of a gas at a constant temperature, as described by Boyle's Law. Similarly, the gravitational force between two objects varies inversely with the square of the distance between them. These examples highlight the practical significance of inverse variation in understanding and modeling various phenomena.

In this specific problem, we are given that the value of m varies inversely as the square of n. This immediately establishes the inverse variation relationship, allowing us to express it mathematically as m n² = k. The problem further provides us with an initial condition: when n = 3, m = 6. This crucial piece of information enables us to determine the constant of variation, k. By substituting these values into the equation, we get:

6 * (3)² = k,

which simplifies to:

6 * 9 = k,

yielding k = 54. Now that we have the constant of variation, k, we can express the specific inverse relationship between m and n as:

m n² = 54.

The problem then asks us to find the positive value of n when m = 13.5. To do this, we substitute m = 13.5 into the equation:

  1. 5 * n² = 54.

To isolate n², we divide both sides of the equation by 13.5:

n² = 54 / 13.5,

which simplifies to:

n² = 4.

To find n, we take the square root of both sides of the equation:

n = ±√4.

This gives us two possible solutions: n = 2 and n = -2. However, the problem specifically asks for the positive value of n, so we discard the negative solution. Therefore, the positive value of n when m = 13.5 is n = 2. This solution aligns with the inverse variation relationship, as an increase in m from 6 to 13.5 corresponds to a decrease in n from 3 to 2, albeit following the inverse square relationship.

Let's break down the solution into a clear, step-by-step process:

  1. Identify the inverse variation relationship: The problem states that m varies inversely as the square of n. This means that m n² = k, where k is the constant of variation.

  2. Determine the constant of variation (k): We are given that when n = 3, m = 6. Substitute these values into the equation m n² = k:

    6 * (3)² = k 6 * 9 = k k = 54

  3. Write the specific inverse variation equation: Now that we know k = 54, we can write the equation as:

    m n² = 54

  4. Substitute the given value of m: We are asked to find n when m = 13.5. Substitute this value into the equation:

    1. 5 * n² = 54

  5. Solve for n²: Divide both sides of the equation by 13.5:

    n² = 54 / 13.5 n² = 4

  6. Solve for n: Take the square root of both sides:

    n = ±√4 n = ±2

  7. Choose the positive value of n: Since the problem asks for the positive value of n, we choose n = 2.

Therefore, the positive value of n when m = 13.5 is 2.

Now, let's analyze the answer choices provided:

A. 4 B. 2 C. 9 D. 18

Based on our step-by-step solution, we determined that the correct answer is n = 2. This corresponds to answer choice B. The other answer choices are incorrect because they do not satisfy the inverse variation relationship established by the problem. To further illustrate this, let's test each answer choice by substituting it back into the equation m n² = 54:

  • Choice A (n = 4): m * (4)² = 54 => m = 54 / 16 = 3.375. This does not match the given m value of 13.5.
  • Choice B (n = 2): m * (2)² = 54 => m = 54 / 4 = 13.5. This matches the given m value, confirming that n = 2 is the correct solution.
  • Choice C (n = 9): m * (9)² = 54 => m = 54 / 81 = 0.666.... This does not match the given m value of 13.5.
  • Choice D (n = 18): m * (18)² = 54 => m = 54 / 324 = 0.1666.... This does not match the given m value of 13.5.

This analysis further solidifies our conclusion that answer choice B (2) is the correct solution.

This problem provides valuable insights into solving inverse variation problems. Here are some key takeaways and general strategies to keep in mind:

  1. Understand the definition of inverse variation: Recognize that when two variables vary inversely, their product is a constant. Mathematically, this is expressed as m n = k or m n² = k, depending on the specific relationship.
  2. Identify the constant of variation: Use the given initial conditions to find the value of the constant of variation (k). This constant is crucial for establishing the specific relationship between the variables.
  3. Write the specific inverse variation equation: Once you have the constant of variation, write the equation that relates the variables. This equation will be the foundation for solving the problem.
  4. Substitute the given values: Substitute the given values into the equation and solve for the unknown variable. Be mindful of the operations you perform to ensure accuracy.
  5. Check your answer: After finding the solution, substitute it back into the original equation to verify that it satisfies the inverse variation relationship. This helps to catch any errors made during the solving process.
  6. Pay attention to the question's requirements: Ensure that you are answering the specific question being asked. For instance, if the question asks for the positive value of a variable, discard any negative solutions.
  7. Practice consistently: The more you practice solving inverse variation problems, the more comfortable you will become with the concept and the solution process.

By mastering these strategies, you will be well-equipped to tackle a wide range of inverse variation problems in mathematics and other disciplines.

In conclusion, the problem presented a classic example of inverse variation, where the value of m varies inversely as the square of n. By understanding the fundamental principles of inverse variation, setting up the equation correctly, and following a step-by-step solution process, we successfully determined that the positive value of n when m = 13.5 is 2. This solution aligns with the inverse variation relationship and is supported by analyzing the answer choices. The key takeaways and general strategies discussed in this article provide a solid foundation for tackling similar problems in the future. Remember to always pay close attention to the problem statement, identify the relevant relationships, and apply the appropriate mathematical techniques to arrive at the correct solution. With consistent practice and a clear understanding of the underlying concepts, you can confidently navigate the world of inverse variation and related mathematical challenges.