Problem 37 - Algebra A Combined Function

Question

For each statement, find the constant of variation and the variation equation. \(y\) varies jointly as \(x\) and the cube of \(z ; y=120\) when \(x=5\) and \(z=2\)

Step-by-Step Solution

Answer

Constant of variation: 3; Equation: \(y = 3xz^3\)

Step 1: Understand Variation Type

The statement "\(y\) varies jointly as \(x\) and the cube of \(z\)" means that \(y\) is directly proportional to the product of \(x\) and \(z^3\). This can be written as the equation \(y = kxz^3\), where \(k\) is the constant of variation.

Step 2: Substitute Given Values into the Equation

We are given that \(y = 120\), \(x = 5\), and \(z = 2\). Substitute these values into the equation \(y = kxz^3\) to find \(k\). This gives us: \[120 = k \times 5 \times 2^3\]

Step 3: Simplify and Solve for Constant of Variation

Calculate \(2^3\) which is \(8\), then substitute back into the equation: \[120 = k \times 5 \times 8\] Simplify the right hand side: \[120 = k \times 40\] Solve for \(k\) by dividing both sides by \(40\): \[k = \frac{120}{40} = 3\]

Step 4: Write the Variation Equation

Substitute the value of \(k\) back into the general equation to get the specific variation equation. The variation equation is: \[y = 3xz^3\]

Key Concepts

Constant of Variation

In the world of mathematics, the constant of variation is a crucial factor when dealing with equations of proportionality. It is a fixed number that relates how two or more variables are connected in a dependency relationship. When one variable changes, the constant ensures that the other variable responds appropriately, preserving the proportion set by the relationship.

For example, in the setup where "\(y\) varies jointly as \(x\) and \(z^3\)", the constant of variation \(k\) simplifies the relationship into the equation \(y = kxz^3\). This equation tells us how \(y\) reacts to changes in \(x\) and \(z\), and \(k\) ensures the relationship remains balanced. More simply, if you know \(x\), \(z\), and \(k\), you can easily find \(y\).

In our exercise, after processing the given values, we found the constant of variation to be \(k = 3\). This number is pivotal as it dictates how strongly \(y\) responds to the changes in \(x\) and \(z^3\). Without this constant, the equation would not maintain its proportional relationship.

Variation Equation

A variation equation is a mathematical expression that outlines how one quantity changes in response to alterations in other variables. The structure of this equation reveals a predictable pattern or "rule" of dependence which is useful in many scientific and economic scenarios.

In the exercise at hand, the variation equation is represented as \(y = kxz^3\), with \(k\) being the constant of variation. This type of equation often appears in problems concerning joint or combined variations — where a variable varies in relation to several other quantities at once.

For our problem, after determining that \(k = 3\), we can express the specific variation equation as \(y = 3xz^3\). This formula specifies that for any changes in \(x\) and \(z\), our outcome \(y\) will adjust based on these inputs, multiplied by our constant \(3\). Thus, it creates a direct path to predict \(y\) once all other factors are known.

Direct Proportionality

Direct proportionality is a core principle in mathematics that describes a specific kind of relationship between two quantities. When we say that \(y\) is directly proportional to a product like \(xz^3\), it means that \(y\) increases or decreases consistently and predictably with \(xz^3\). This predictable change is maintained by the constant of variation.

To illustrate, suppose you double \(x\) while keeping \(z\) constant, the value of \(y\) will also double, assuming the constant \(k\) remains unchanged. The equation \(y = kxz^3\), therefore, reflects this direct proportionality, ensuring the relationship between the variables does not falter. This steady transformation is why direct proportionality is such a valuable concept in solving real-world problems, whether in physics, chemistry, or economics.

In our example, by knowing \(y, x, z\), and \(k\), we maintain this direct proportionality and ensure that all values remain in balance, showing clearly how one variable impacts another.