Problem 19 - College Algebra

Question

For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies jointly as \(x\) and the square root of \(z\) and when \(x=2\) and \(z=25,\) then \(y=100\).

Step-by-Step Solution

Answer

The equation is \( y = 10x \sqrt{z} \).

Step 1: Understand the Joint Variation

When a variable varies jointly with other variables, it means that it is directly proportional to the product of those variables. In this case, \( y \) varies jointly as \( x \) and the square root of \( z \), which means we can write this relationship as \( y = kx \sqrt{z} \), where \( k \) is the constant of proportionality.

Step 2: Substitute Given Values to Find \( k \)

We know \( y = 100 \), \( x = 2 \), and \( z = 25 \). Substitute these values into the equation \( y = kx \sqrt{z} \) to find \( k \). \[ 100 = k \times 2 \times \sqrt{25} \] Simplifying the square root, \( \sqrt{25} = 5 \), so: \[ 100 = k \times 2 \times 5 \] \[ 100 = 10k \].

Step 3: Solve for \( k \)

Now divide both sides of the equation by 10 to solve for \( k \): \[ k = \frac{100}{10} \] \[ k = 10 \].

Step 4: Write the Equation

Now that we have found \( k = 10 \), substitute \( k \) back into the relationship to formulate the complete equation: \[ y = 10x \sqrt{z} \]. This equation describes \( y \) as it varies jointly with \( x \) and the square root of \( z \).

Key Concepts

Constant of Proportionality

When dealing with variations, the **constant of proportionality** is a critical factor that connects the relationship between variables. It helps indicate how much one variable changes concerning others. In our example, the constant of proportionality, denoted by **k**, shows how the variable **y** varies in response to changes in **x** and the square root of **z**.

To find the constant of proportionality, we use given specific values to solve for **k**. In this exercise, when given values are substituted into the joint variation equation \( y = kx \sqrt{z} \), we used \( y = 100, x = 2, \text{ and } z = 25 \) to determine \( k \). Hence, \( k \) = 10 in this case.

• **Proportionality** indicates a stable relationship.
• **Constant** implies it does not change with these specific conditions.
• We use it to predict other variables' behavior consistently.

Direct Variation

Understanding **direct variation** is vital in many algebraic relationships. This concept signifies that two variables increase or decrease proportionally, at a steady rate. It means when one variable increases, the other follows suit. You can describe it with equations like \( y = kx \), showing a direct link.

In the context of this problem, starting with the equation \( y = kx \sqrt{z} \), the variable **y** increases directly as \( x \) and the square root of **z** increase. Here, our task is to see how changes in \( x \) and \( z \) directly affect **y**.

• Summarized as linear and straightforward.
• Chained together by a **constant (k)**.
• Predictability in one variable from another's change.

Square Root

Unraveling the **square root** concept deals with understanding the fundamental operation of finding a number that, when multiplied by itself, returns the original value. For instance, the square root of 25 is 5 because \( 5 \times 5 = 25 \).

In the given exercise, the square root of **z** plays a part in how **y** varies with both **x** and \( \sqrt{z} \). By taking the square root of **z**, we modify how substantially **z** contributes to the outcome. It is crucial for balancing the equation and understanding how **z** influences **y**.

• Provides a way to simplify complex relationships.
• Alters how much a variable contributes to the solution.
• Key in reducing or balancing in multi-variable scenarios.

Algebraic Equation

An **algebraic equation** represents mathematical relationships using symbols and numbers. In these equations, equality is expressed through one or more variables.

In this context, the algebraic equation provided is **\( y = kx \sqrt{z} \)**. It describes a joint variation where changes in **x** and \( \sqrt{z} \) lead to changes in **y**. Each component plays a specific role in solving the equation, ensuring both sides equal as alterations in the variables control the outcome. Finding **k** solidifies the relationship, making the equation both specific and useful.

• Serves as a structured mathematical statement.
• Important in expressing predictable behavior in varied conditions.
• Ensures logical solutions for diverse problems.