Problem 6 - College Algebra

Question

For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies directly as the square root of \(x\) and when \(x=36, y=24\).

Step-by-Step Solution

Answer

The equation is \( y = 4 \sqrt{x} \).

Step 1: Understand direct variation

When we say that a variable \( y \) varies directly as the square root of \( x \), it means we can express their relationship using the equation \( y = k \sqrt{x} \), where \( k \) is a constant. Our task is to find the value of the constant \( k \).

Step 2: Substitute known values

We are given that when \( x = 36 \), \( y = 24 \). Substitute these values into the equation \( y = k \sqrt{x} \) to determine \( k \). So, \( 24 = k \sqrt{36} \).

Step 3: Calculate \( \sqrt{x} \)

Find the square root of \( x \). In this case, \( \sqrt{36} = 6 \).

Step 4: Solve for \( k \)

Substitute \( \sqrt{36} = 6 \) into the equation from Step 2: \( 24 = k (6) \). Divide both sides by 6 to solve for \( k \). This gives \( k = \frac{24}{6} = 4 \).

Step 5: Write the final equation

Now that we have \( k \), substitute it back into the equation \( y = k \sqrt{x} \). The relationship between \( y \) and \( x \) is therefore \( y = 4 \sqrt{x} \).

Key Concepts

Square Root

The concept of a square root is fundamental in algebra. When we talk about the square root of a number, we're referring to a value that, when multiplied by itself, gives the original number. For example, the square root of 36 is 6 because 6 multiplied by itself (6 × 6) equals 36. This operation helps in simplifying equations and understanding relationships between variables.

• Notion: The square root is represented by the radical symbol (√).
• Use in equations: Solves many algebraic problems by simplifying numbers under the radical.
• Relation to direct variation: Shows how a variable can directly change as another's square root changes.

In direct variation problems, understanding square roots allows us to describe how one variable changes with respect to another. This is crucial when forming and solving equations.

Constant of Variation

The constant of variation, represented by the symbol \( k \), is a key component in direct variation equations. It serves as the proportionality constant that describes how much one variable changes with respect to a change in another.

• Definition: \( k \) is the multiplier that scales the relationship between variables.
• Role: Determines how steep or shallow this variation is.
• Dependence: On the initial conditions or values given in the problem.

In the context of direct variation like \( y = k \sqrt{x} \), \( k \) tells us the rate at which \( y \) changes with respect to changes in the square root of \( x \). This serves to tailor the basic variation formula to accurately model the given data.

Algebraic Equation

An algebraic equation is a mathematical statement that shows the equality between two expressions. These equations are essential in solving for unknown values and understanding relationships between variables.

• Structure: Composed of variables, constants, and operations such as addition, multiplication, and roots.
• Purpose: To express relationships and solve for unknowns.
• Example: In our exercise, the equation \( y = k \sqrt{x} \) shows the direct relationship between \( y \) and the square root of \( x \).

In practice, algebraic equations like the one above are used to model real-world situations. They allow students to translate word problems into mathematical statements that can be manipulated to find solutions.

Solving for k

To find the constant of variation \( k \) in an algebraic equation, like \( y = k \sqrt{x} \), the process involves substituting known values and algebraic manipulation.

• Substitute given values into the equation to replace variables with numbers.
• Calculate necessary components, like square roots, before solving for \( k \).
• Isolate \( k \) by performing inverse operations.

In our problem where \( x = 36 \) and \( y = 24 \), you substitute these values into the equation to get \( 24 = k (6) \). Solving this equation by dividing both sides by 6 gives \( k = 4 \). This process isolates \( k \) and gives you the exact proportion necessary to maintain the direct variation described.