Problem 43 - Intermediate Algebra

Question

If \(y\) is inversely proportional to the square root of \(x\), and if \(y=0.08\) when \(x=225\), find \(y\) when \(x=625\).

Step-by-Step Solution

Answer

When \( x = 625 \), \( y = 0.048 \).

Step 1: Understand the relationship

Since \( y \) is inversely proportional to the square root of \( x \), we can express this relationship as \( y = \frac{k}{\sqrt{x}} \), where \( k \) is a constant. Our task is to find \( y \) for \( x = 625 \), given that \( y = 0.08 \) when \( x = 225 \).

Step 2: Find the constant \( k \)

We use the given values to find \( k \). Substituting \( y = 0.08 \) and \( x = 225 \) into the equation, we get: \[ 0.08 = \frac{k}{\sqrt{225}} \] Since \( \sqrt{225} = 15 \), this simplifies to: \[ 0.08 = \frac{k}{15} \] Solving for \( k \), we multiply both sides by 15: \[ k = 0.08 \times 15 = 1.2 \] So, the constant \( k \) is \( 1.2 \).

Step 3: Calculate \( y \) for \( x = 625 \)

Now we use the constant \( k = 1.2 \) and the new value \( x = 625 \) to find \( y \). Plug these into the equation \( y = \frac{k}{\sqrt{x}} \): \[ y = \frac{1.2}{\sqrt{625}} \] Since \( \sqrt{625} = 25 \), this becomes: \[ y = \frac{1.2}{25} \] Divide 1.2 by 25 to find \( y \): \[ y = 0.048 \] Thus, when \( x = 625 \), \( y = 0.048 \).

Key Concepts

Square Root

When dealing with inverse proportionality and square roots, it's crucial to grasp what a square root is and its use in equations. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 225 is 15, because \( 15 \times 15 = 225 \). Similarly, for 625, the square root is 25, because \( 25 \times 25 = 625 \).
Understanding square roots helps solve problems that involve inverse proportionality. By knowing that \( y \) is inversely proportional to the square root of \( x \), we express it as \( y = \frac{k}{\sqrt{x}} \). This equation makes square root calculations essential for finding either the constant of proportionality or another variable.

• Calculate the square root for given values to understand changes in the proportionality.
• Recognize that square roots simplify the multiplication required in our equations.

Developing a firm understanding of square roots is a fundamental step in mastering equations involving proportions and ratios.

Constant of Proportionality

The constant of proportionality, often denoted as \( k \), is a constant value that describes the relationship between variables in proportional equations. In our inverse proportionality equation \( y = \frac{k}{\sqrt{x}} \), \( k \) indicates how \( y \) changes with variations in \( x \).
To determine \( k \), use known values for \( y \) and \( x \). From our exercise, \( y = 0.08 \) when \( x = 225 \). Substituting these into the equation, we solve for \( k \):

• Find \( \sqrt{225} = 15 \).
• Set up the equation \( 0.08 = \frac{k}{15} \).
• Multiply both sides by 15 to isolate \( k \): \( k = 0.08 \times 15 = 1.2 \).

Having the constant \( k \), you can predict how \( y \) changes when \( x \) changes, ensuring accuracy and consistency in proportional relationships.

Substitution Method

The substitution method is a technique used to solve equations by substituting known values into an equation to find an unknown variable. In our example, once the constant \( k \) was determined as 1.2, substitution was key to finding \( y \) for a different \( x \) value.
Here's how substitution works:

• First, identify equations and known values. We've established \( y = \frac{k}{\sqrt{x}} \) and \( k = 1.2 \).
• Substitute the new \( x \) value, 625, into the equation to find \( y \).
• Calculate \( \sqrt{625} = 25 \).
• Replace the values into \( y = \frac{1.2}{25} \).
• Solving gives \( y = 0.048 \).

Substitution is invaluable in algebra, making it easier to handle complex equations by reducing them into simpler computations with known quantities. It's efficient and perfect for equations involving direct or inverse relationships.