Solving Variation Problems Find K And Z When Y Varies Directly As X And Inversely As Z
This problem delves into the fundamental concepts of direct and inverse variation, crucial in mathematical problem-solving. Understanding these relationships allows us to model and predict how quantities change in relation to one another. In this article, we will explore the given scenario where $y$ varies directly as $x$ and inversely as $z$. We'll dissect the problem step-by-step, guiding you through finding the constant of variation and solving for specific values. Let's embark on this journey to unravel the intricacies of variation and empower you with the skills to tackle similar problems confidently.
(a) Finding the Value of $K$, the Constant of Variation
To begin, let's define what it means for $y$ to vary directly as $x$ and inversely as $z$. Direct variation implies that as $x$ increases, $y$ increases proportionally, and vice versa. Mathematically, this is represented as $y = kx$, where $k$ is the constant of variation. Inverse variation, on the other hand, means that as $z$ increases, $y$ decreases proportionally, and vice versa. This relationship is expressed as $y = \frac{k}{z}$, where $k$ is again the constant of variation. When both direct and inverse variations are present, we combine these relationships into a single equation. In this case, since $y$ varies directly as $x$ and inversely as $z$, we can write the equation as:
Here, $K$ represents the constant of variation that ties the relationship between $x$, $y$, and $z$. Our first task is to determine the value of this constant. We are given a set of values for $x$, $y$, and $z$: $x = 2$, $y = 4$, and $z = 3$. We can substitute these values into the equation to solve for $K$:
To isolate $K$, we need to get rid of the fraction. We can do this by multiplying both sides of the equation by $\frac{3}{2}$:
Simplifying the equation, we get:
Therefore, the value of $K$, the constant of variation, is 6. This constant is crucial as it defines the specific relationship between $x$, $y$, and $z$ in this scenario. With the value of $K$ determined, we can now rewrite our equation as:
This equation allows us to find the value of any one variable if we know the values of the other two. It encapsulates the direct relationship between $y$ and $x$, the inverse relationship between $y$ and $z$, and the specific constant of proportionality that governs their interaction. In the next part of the problem, we will use this equation to find the value of $z$ when given new values for $x$ and $y$.
(b) Finding $z$ when $x = 12$ and $y = 10$
Now that we have determined the constant of variation, $K = 6$, we can move on to the second part of the problem: finding the value of $z$ when $x = 12$ and $y = 10$. We will use the equation we derived earlier:
We are given $x = 12$ and $y = 10$. Substituting these values into the equation, we get:
Our goal is to isolate $z$. First, we can simplify the right side of the equation by multiplying 6 and 12:
Now, to get $z$ out of the denominator, we can multiply both sides of the equation by $z$:
Finally, to solve for $z$, we divide both sides of the equation by 10:
Simplifying the fraction, we get:
Therefore, when $x = 12$ and $y = 10$, the value of $z$ is 7.2. This result demonstrates how the inverse variation between $y$ and $z$ plays out. As $x$ increases (from 2 to 12) and $y$ also increases (we've set it to 10), $z$ must adjust to maintain the balance dictated by the constant of variation. This highlights the interconnectedness of the variables and how they influence each other within the system defined by the variation equation.
In summary, we have successfully navigated the problem by first finding the constant of variation, which defines the specific relationship between the variables, and then using that constant to solve for an unknown variable given the values of the others. This process underscores the power of understanding direct and inverse variation in solving mathematical problems and modeling real-world scenarios.
Conclusion: Mastering Direct and Inverse Variation
This problem has provided a valuable exercise in understanding and applying the concepts of direct and inverse variation. By systematically working through each part, we have reinforced the importance of these relationships in mathematical problem-solving. We began by defining direct and inverse variation, then combined these concepts to form the equation $y = K \frac{x}{z}$. We successfully found the constant of variation, $K$, by substituting the given values and solving for the unknown. This constant acts as a key that unlocks the specific relationship between the variables in this scenario.
Next, we applied our newfound knowledge to find the value of $z$ when given new values for $x$ and $y$. This step demonstrated how to manipulate the equation and isolate the desired variable. The solution highlighted the interplay between the variables and how they adjust in response to changes in each other, governed by the constant of variation. This underscores the importance of a systematic approach to problem-solving, starting with understanding the underlying concepts and then applying them methodically.
Furthermore, this exercise exemplifies the practical applications of mathematical concepts. Direct and inverse variation are not just abstract ideas; they are powerful tools for modeling real-world phenomena. From physics to economics, these relationships are used to describe how quantities change in response to one another. Understanding these concepts allows us to make predictions, solve problems, and gain a deeper understanding of the world around us. To solidify your understanding, consider practicing more problems involving direct and inverse variation. Look for patterns, explore different scenarios, and challenge yourself to apply these concepts in novel ways. The more you practice, the more confident you will become in your ability to tackle these types of problems.
In conclusion, mastering direct and inverse variation opens doors to a broader understanding of mathematics and its applications. It empowers you to analyze relationships between quantities, solve problems systematically, and make informed decisions based on mathematical principles. Embrace these concepts, practice diligently, and unlock your full potential in the world of mathematics.
