Solving Joint Variation Problems Find Y When X Is 1 And W Is 5

In the realm of mathematics, joint variation problems often appear, requiring a clear understanding of the relationships between variables. This article delves into a specific joint variation problem, providing a step-by-step solution and shedding light on the underlying principles. We will explore how to find the value of y when it varies jointly as x and w, given an initial set of values and a new set of conditions. This exploration will not only provide the answer to the specific problem but also equip you with the knowledge to tackle similar joint variation scenarios.

Understanding Joint Variation

In joint variation problems, a variable varies directly as the product of two or more other variables. In simpler terms, if one variable increases, the other variables also increase proportionally, and vice versa. The core concept lies in the constant of variation, which dictates the proportionality between the variables. To truly grasp the solution, it's essential to understand the fundamental concept of joint variation. Joint variation occurs when a variable varies directly as the product of two or more other variables. This means that the variable in question changes proportionally with the product of the other variables. The mathematical representation of this relationship involves a constant of variation, which plays a crucial role in solving joint variation problems. This constant represents the fixed ratio between the variable and the product of the other variables. Understanding the constant of variation is key to setting up and solving the equation that represents the joint variation relationship. It allows us to determine how the variable changes when the other variables change, and it forms the foundation for finding the unknown value in the problem. Without a solid grasp of joint variation, tackling these problems can feel like navigating a maze. The key is to recognize the direct proportionality between the variable in question and the product of the other variables, and to understand the role of the constant of variation in maintaining this proportionality. By mastering this concept, you'll be well-equipped to solve a wide range of joint variation problems, from simple scenarios to more complex applications in various fields.

Setting up the Equation

To solve joint variation problems, we must translate the given information into a mathematical equation. This equation will express the relationship between the variables and the constant of variation. The general form of the equation is y = kxw, where y varies jointly as x and w, and k is the constant of variation. This equation serves as the foundation for solving the problem. The first step is to identify the variables involved and their relationship as described in the problem statement. In this case, we have three variables: y, x, and w, and we are told that y varies jointly as x and w. This means that y is directly proportional to the product of x and w. To express this relationship mathematically, we introduce the constant of variation, k, which represents the factor by which the product of x and w must be multiplied to obtain y. The equation y = kxw captures this relationship precisely. It states that y is equal to the product of k, x, and w. This equation is the cornerstone of the solution, and it will be used to find the value of y under different conditions. By understanding how to set up this equation, you'll be able to approach any joint variation problem with confidence. Remember that the constant of variation, k, remains constant throughout the problem, and it is this constant that links the variables together in a predictable and proportional manner. This foundational equation is not just a formula; it's a representation of the underlying mathematical relationship between the variables, and mastering its setup is essential for solving joint variation problems.

Problem Statement: y Varies Jointly as x and w

Our problem states that y varies jointly as x and w. This means that y is directly proportional to the product of x and w. We are given that y = 48 when x = 6 and w = 2. Our goal is to find the value of y when x = 1 and w = 5. To reiterate, the problem provides us with a specific scenario of joint variation. We know that y varies jointly as x and w, which means that y is directly proportional to the product of x and w. This relationship is the key to solving the problem. The problem also gives us an initial set of values for the variables: y = 48 when x = 6 and w = 2. These values will be crucial in determining the constant of variation, which is the fixed ratio between y and the product of x and w. Once we find the constant of variation, we can use it to find the value of y when x = 1 and w = 5. This is the ultimate goal of the problem. The problem statement is concise and clear, providing all the necessary information to solve it. The challenge lies in translating the words into mathematical equations and then using those equations to find the unknown value. By carefully analyzing the problem statement and identifying the given information and the desired outcome, we can develop a strategic approach to solve the problem. The problem's structure is typical of joint variation problems, and the skills we develop in solving it will be applicable to a wide range of similar problems.

Step 1: Finding the Constant of Variation (k)

To find the constant of variation (k), we substitute the given values (y = 48, x = 6, w = 2) into the equation y = kxw. This gives us 48 = k * 6 * 2. Simplifying this equation, we get 48 = 12k. Dividing both sides by 12, we find that k = 4. The constant of variation, k, is a crucial element in joint variation problems. It represents the fixed ratio between the dependent variable (y in this case) and the product of the independent variables (x and w). Finding the value of k is the first essential step in solving the problem. We achieve this by using the initial set of values provided in the problem statement: y = 48 when x = 6 and w = 2. These values allow us to establish a specific relationship between the variables and the constant of variation. Substituting these values into the equation y = kxw gives us 48 = k * 6 * 2. This equation is a simple algebraic equation with one unknown, k. Solving for k involves simplifying the equation and isolating k on one side. We first multiply 6 and 2 to get 12, resulting in 48 = 12k. Then, we divide both sides of the equation by 12 to solve for k. This gives us k = 4. The value of k, 4, represents the constant of variation in this specific problem. This means that y is always 4 times the product of x and w. This constant is the key to unlocking the solution for any other set of values for x and w. By finding the constant of variation, we have established the fundamental relationship between the variables, and we are now ready to proceed to the next step: finding the value of y when x = 1 and w = 5.

Step 2: Calculating y when x = 1 and w = 5

Now that we have found the constant of variation (k = 4), we can use it to find y when x = 1 and w = 5. Substituting these values into the equation y = kxw, we get y = 4 * 1 * 5. This simplifies to y = 20. Therefore, when x = 1 and w = 5, y = 20. With the constant of variation, k = 4, firmly established, we can now tackle the primary objective of the problem: finding the value of y when x = 1 and w = 5. This is where the power of joint variation becomes evident. We simply substitute these new values of x and w, along with the value of k, into the equation y = kxw. This gives us y = 4 * 1 * 5. This equation is a straightforward multiplication problem. We multiply 4, 1, and 5 together to find the value of y. The multiplication is simple: 4 multiplied by 1 is 4, and then 4 multiplied by 5 is 20. Therefore, y = 20. This is the solution to the problem. When x = 1 and w = 5, the value of y is 20. This result demonstrates the direct proportionality between y and the product of x and w, as defined by the constant of variation. The constant of variation acts as a bridge, linking the variables together and allowing us to predict the value of y for any given values of x and w. By finding the constant of variation and then using it to calculate y, we have successfully solved the joint variation problem. This process highlights the importance of understanding the relationship between variables and the constant of variation in joint variation scenarios. The solution y = 20 is not just a numerical answer; it's a testament to the power of mathematical relationships and the ability to use them to solve real-world problems.

Answer: D. 20

Therefore, the correct answer is D. 20. This option aligns with our calculated value of y when x = 1 and w = 5. This option, D. 20, is the culmination of our step-by-step solution. It represents the precise value of y when x = 1 and w = 5, based on the joint variation relationship defined in the problem. The fact that this answer matches our calculated value reinforces the accuracy of our solution process. We began by understanding the concept of joint variation, setting up the equation y = kxw, and then finding the constant of variation, k. This constant served as the crucial link between the variables, allowing us to predict the value of y for any given values of x and w. With k firmly established, we substituted the new values of x and w into the equation and calculated y. The result, y = 20, is the final answer, and it aligns perfectly with option D. This agreement between our calculated result and the provided option is a testament to the power of mathematical reasoning and the importance of following a systematic approach to problem-solving. The process we followed not only led us to the correct answer but also provided a deeper understanding of joint variation and its applications. The answer D. 20 is not just a number; it's a representation of the mathematical relationship between the variables and the constant of variation in this specific scenario. It showcases the ability to translate a verbal problem into a mathematical equation and then use that equation to find a precise solution.

Conclusion

In conclusion, we have successfully solved the joint variation problem by finding the constant of variation and using it to calculate the value of y when x = 1 and w = 5. The answer is 20. We've navigated the intricacies of joint variation, from understanding its fundamental principles to applying them in a practical problem-solving scenario. The journey began with recognizing the joint variation relationship, where y varies directly as the product of x and w. This understanding formed the foundation for setting up the equation y = kxw, which represents the mathematical relationship between the variables and the constant of variation, k. The first crucial step was to find the value of k, the constant of variation. We achieved this by substituting the initial values given in the problem statement (y = 48, x = 6, w = 2) into the equation. This allowed us to solve for k, which we found to be 4. With k firmly established, we moved on to the primary objective: finding the value of y when x = 1 and w = 5. We simply substituted these new values, along with the value of k, into the equation y = kxw. The resulting calculation led us to the solution: y = 20. This answer confirms the direct proportionality between y and the product of x and w, as dictated by the constant of variation. The entire process highlights the power of mathematical reasoning and the importance of a systematic approach to problem-solving. By breaking down the problem into manageable steps, we were able to identify the key concepts, apply the appropriate equations, and arrive at the correct answer. The solution, y = 20, is not just a numerical result; it's a testament to the ability of mathematics to model real-world relationships and provide precise answers to complex problems. This problem-solving experience has equipped us with the skills and understanding to tackle a wide range of joint variation problems in the future. The knowledge gained extends beyond the specific problem and provides a valuable framework for approaching similar mathematical challenges.