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Chapter 3: Problem 642

For the following exercises, use the given information to find the unknown value. \(y\) varies jointly as \(x\) and \(z .\) When \(x=4\) and \(z=2,\) then \(y=16\) . Find \(y\) when \(x=3\) and \(z=3\).

Short Answer

Expert verified
When \(x=3\) and \(z=3\), \(y=18\).

Step by step solution

01

Understand the Concept of Joint Variation

When a variable varies jointly as two other variables, it means that the variable depends directly on the product of the two other variables. For this problem, if \(y\) varies jointly as \(x\) and \(z\), we can express this relationship as \(y = kxz\), where \(k\) is the constant of variation.
02

Determine the Constant of Variation

We know that when \(x = 4\), \(z = 2\), then \(y = 16\). We can use these values to determine \(k\) in the equation \(y = kxz\). Substitute the given values to solve for \(k\):\[16 = k \cdot 4 \cdot 2\]Simplify to find \(k\):\[16 = 8k\]\[k = 2\]
03

Use the Constant to Find the Unknown Value of y

Now that \(k\) is known, use it to find \(y\) when \(x = 3\) and \(z = 3\). Substitute \(x = 3\), \(z = 3\), and \(k = 2\) into the equation \(y = kxz\):\[y = 2 \cdot 3 \cdot 3\]Calculate \(y\):\[y = 18\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Constant of Variation
The constant of variation is a key element in understanding algebraic relationships, especially in problems involving joint variation. In the problem above, the constant of variation is represented by the letter \(k\). This constant is what links the variables in the equation together. It's essentially a fixed number that describes how the variables interact with one another.
  • When the relationship among variables is described, like in \(y = kxz\), the constant of variation \(k\) makes it possible to connect \(y\) with both \(x\) and \(z\).
  • In the given exercise, by substituting the known values for \(y\), \(x\), and \(z\), we calculate \(k = 2\). This remains the same regardless of what particular values \(x\) and \(z\) take, as long as the direct relationship holds true.
Knowing the constant of variation allows you to predict or calculate any of the variables, provided you have enough information about the others.
Direct Variation
Direct variation is a specific type of algebraic relationship where one variable increases or decreases directly as another variable does. In our earlier scenario, \(y\) directly varies with the product of \(x\) and \(z\). This is expressed in the equation \(y = kxz\). When describing direct variation:
  • If \(x\) or \(z\) increases, \(y\) will also increase if \(k\) is positive.
  • This is because all changes are multiplied by the constant \(k\), ensuring consistent proportional scaling.
Direct variation provides a simple, yet powerful way to understand how different quantities are interrelated mathematically. It relies on maintaining a consistent ratio defined by the constant of variation.
Algebraic Relationships
Algebraic relationships help to describe how variables are related to each other using mathematical expressions and equations. In the context of the exercise, we encounter a joint variation, which is a specialized algebraic relationship. Here's how it works:
  • Joint variation involves one variable (in this case, \(y\)) varying directly with the product of two other variables, forming the equation \(y = kxz\).
  • This type of relationship is essentially a blend of direct variations involving multiple factors.
  • By manipulating the equation, we can determine the value of one variable if the other variables and the constant of variation are known, showcasing the power of algebra to model real-world situations.
Algebraic relationships like these form the backbone of many mathematical applications, illustrating how different quantities behave in tandem.

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