Problem 20 - College Algebra with Corequisite Support

Question

For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies jointly as the square of \(x\) the cube of \(z\) and the square root of \(W\). When \(x=1, z=2\), and \(w=36,\) then \(y=48\).

Step-by-Step Solution

Answer

The equation is \(y = x^2 \cdot z^3 \cdot \sqrt{W}\).

Step 1: Understanding Joint Variation

Joint variation means that the variable of interest, in this case, \(y\), depends on two or more variables. Specifically, \(y\) varies jointly as the square of \(x\), the cube of \(z\), and the square root of \(W\). This can be mathematically expressed as \(y = k \cdot x^2 \cdot z^3 \cdot \sqrt{W}\), where \(k\) is the constant of proportionality.

Step 2: Substituting Known Values

To find the constant \(k\), substitute the given known values into the equation: \(x = 1\), \(z = 2\), \(W = 36\), and \(y = 48\). The equation becomes: \[ 48 = k \cdot 1^2 \cdot 2^3 \cdot \sqrt{36} \] Simplify the terms to find the value of \(k\).

Step 3: Simplifying the Equation

Simplify the known values in the equation: - \(1^2 = 1\)- \(2^3 = 8\)- \(\sqrt{36} = 6\)So the equation becomes: \[ 48 = k \cdot 1 \cdot 8 \cdot 6 \] This further simplifies to: \[ 48 = 48k \]

Step 4: Solving for the Constant

Divide both sides of the equation by 48 to isolate \(k\): \[ k = \frac{48}{48} = 1 \] Thus, the constant of proportionality \(k\) is 1.

Step 5: Writing the Final Equation

Substitute the constant \(k = 1\) back into the original joint variation equation. Therefore, the final equation describing the relationship of the variables is:\[ y = x^2 \cdot z^3 \cdot \sqrt{W} \]

Key Concepts

Constant of Proportionality

In the realm of joint variation, the constant of proportionality, often denoted as \( k \), is crucial. This constant acts as the bridge connecting several variables in a mathematical relationship. In our exercise, we see \( y \) depends on \( x^2 \), \( z^3 \), and \( \sqrt{W} \). The equation is formulated as \( y = k \cdot x^2 \cdot z^3 \cdot \sqrt{W} \). Here, \( k \) represents the adjustment factor that balances the relationship so the equation holds true under specific conditions.
To determine \( k \), you substitute known values into the equation and solve for \( k \). In this case, with \( y = 48 \), \( x = 1 \), \( z = 2 \), and \( W = 36 \), we found \( k = 1 \). This reveals that the sum of the influences of \( x^2 \), \( z^3 \), and \( \sqrt{W} \) directly equals \( y \) without any need for scaling, hence \( k \) is 1.

Mathematical Expression

Mathematical expressions are structured forms combining numbers, variables, and operations. In joint variation, expressions help depict how multiple factors affect a single variable. The expression \( y = k \cdot x^2 \cdot z^3 \cdot \sqrt{W} \) shows how \( y \) varies with changes in \( x \), \( z \), and \( W \).
In our exercise, the expression incorporates:

• \( x^2 \): The square of \( x \) representing its quadratic influence on \( y \).
• \( z^3 \): The cubic power of \( z \) demonstrating a more substantial effect than \( x \).
• \( \sqrt{W} \): The square root of \( W \), indicating a proportional increase in \( y \) as \( W \) grows.

These terms are multiplied together, with the constant \( k \) adjusting their collective effect to equal \( y \). It shows a clear path for understanding how alterations in \( x \), \( z \), and \( W \) shift \( y \).

Equation Writing

When writing an equation that encapsulates joint variation, start by identifying the relationships involved. This includes acknowledging how each variable affects the main variable, \( y \). For joint variation exercises, like the one at hand, it is essential to consider:

• The operation on each variable: For \( x \), \( x^2 \) shows a squared effect. \( z^3 \) reflects cubed influence, and \( \sqrt{W} \) applies a square root transformation.
• The constant of proportionality \( k \): Initially unknown, clarified once the expression is resolved with given values.

Steps for writing the equation:
1. Draft the core relationship: \( y = k \cdot x^2 \cdot z^3 \cdot \sqrt{W} \)
2. Substitute known values to solve for \( k \).
3. Finalize the equation by including \( k \).

This process results in an easily interpretable equation that captures the interplay of multiple variables with \( y \). For our exercise, this was accomplished by substituting and solving, culminating in the consolidated formula: \( y = x^2 \cdot z^3 \cdot \sqrt{W} \).