Problem 39 - College Algebra with Corequisite Support

Question

For the following exercises, use the given information to find the unknown value. \(y\) varies jointly as the square of \(x\) and the cube of \(z\) and inversely as the square root of \(w\). When \(x=2, z=2,\) and \(w=64,\) then \(y=12 .\) Find \(y\) when \(x=1, z=3,\) and \(w=4\).

Step-by-Step Solution

Answer

The value of \( y \) is 40.5.

Step 1: Understand the Relationship

The problem states that \( y \) varies jointly as the square of \( x \), the cube of \( z \), and inversely as the square root of \( w \). Therefore, the formula can be expressed as: \[ y = k \cdot \frac{x^2 \cdot z^3}{\sqrt{w}} \] where \( k \) is the constant of variation.

Step 2: Find the Constant of Variation, k

Use the given values \( x=2, z=2, w=64, \text{ and } y=12 \) to find \( k \). Substitute these values into the equation: \[ 12 = k \cdot \frac{2^2 \cdot 2^3}{\sqrt{64}} \]Calculate the expression: \[ 12 = k \cdot \frac{4 \cdot 8}{8} \]\[ 12 = k \cdot 4 \]Solve for \( k \): \[ k = \frac{12}{4} = 3 \].

Step 3: Use the Constant to Find New y

Now that we have \( k = 3 \), use it to find the new value of \( y \) when \( x=1, z=3, \text{ and } w=4 \). Substitute these values and \( k \) into the expression:\[ y = 3 \cdot \frac{1^2 \cdot 3^3}{\sqrt{4}} \] Calculate the expression:\[ y = 3 \cdot \frac{1 \cdot 27}{2} \]\[ y = 3 \cdot 13.5 \]\[ y = 40.5 \].

Key Concepts

Constant of Variation

The journey begins with understanding the constant of variation — a key component in joint variation topics. In mathematics, the constant of variation, denoted as \( k \), is a fixed number that relates all the varying elements in a formula. In our exercise, the variable \( y \) is defined by how \( x \), \( z \), and \( w \) interact through \( k \). This interaction is given by the formula:

• Jointly as the square of \( x \) and the cube of \( z \): Meaning \( x^2 \cdot z^3 \) will be part of the relationship.
• Inversely as the square root of \( w \): Inverted relationships in algebra express how one variable decreases as another increases, represented here as \( \frac{1}{\sqrt{w}} \).

Combine these relationships using \( k \) to create:
\[ y = k \cdot \frac{x^2 \cdot z^3}{\sqrt{w}} \]Once you've established \( k \) using given conditions, this constant applies to any such situation described in the problem, allowing you to find unknowns effectively.

Word Problems in Algebra

Word problems in algebra can initially seem daunting, but breaking them into manageable parts can make them much easier to handle. The key is to uncover the underlying mathematical relationships behind the words. In our specific exercise, extracting the functional relationship between variables is crucial:

• Identify what varies and how: Here, \( y \) varies jointly and inversely based on certain conditions.
• Translate the variation description into a mathematical formula.
• Use given values to determine any constants or solve for unknown quantities.

Practice makes perfect with word problems. Each problem helps sharpen skills in translating real-world scenarios into mathematical language, preparing you for similar challenges in the future.

Power and Root Relationships

Understanding power and root relationships is fundamental in mastering algebraic expressions and variations.
In our exercise, several power and root components emerge: both \( x \) and \( z \) include power relationships, specifically the square of \( x \) and the cube of \( z \). Powers show how a number can be multiplied by itself a certain number of times, providing a quick way to handle repetitive multiplication.

• Power of 2: \( x^2 = x \times x \)
• Power of 3: \( z^3 = z \times z \times z \)

Roots, like the square root in \( \sqrt{w} \), serve as the inverse operation to powers. They supply a number that, when multiplied by itself, returns the original quantity. Together, powers and roots form a balancing act. They help detail complex relationships in algebraic equations by simplifying and quantifying them, which is crucial in calculating variations like those presented in joint variation problems.