Problem 14 - College Algebra with Corequisite Support

Question

For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies inversely as the square root of \(x\) and when \(x=25, \quad y=3\).

Step-by-Step Solution

Answer

The equation is \( y = \frac{15}{\sqrt{x}} \).

Step 1: Understanding Inverse Variation

In inverse variation, one variable increases while the other decreases, such that their product remains constant. If \(y\) varies inversely as the square root of \(x\), it implies that \(y \times \sqrt{x} = k\), where \(k\) is a constant.

Step 2: Setting up the Equation

The relationship given can be expressed as: \[ y = \frac{k}{\sqrt{x}} \]Where \(k\) is a constant that we need to determine.

Step 3: Solving for the Constant

We use the information given, \(x = 25\) and \(y = 3\), to solve for the constant \(k\). Substituting these values in the inverse variation equation, we have: \[ 3 = \frac{k}{\sqrt{25}} \]\[ 3 = \frac{k}{5} \]Multiply both sides by 5 to solve for \(k\):\[ k = 15 \]

Step 4: Writing the Equation

Now that we have determined the constant \(k\), we can write the complete equation describing the relationship: \[ y = \frac{15}{\sqrt{x}} \]

Key Concepts

Square Root Functions

Square root functions involve the square root of a number or variable. In our example, the square root of the variable \(x\) comes into play. The basic form of a square root function is \(f(x) = \sqrt{x}\).

• The square root symbol \(\sqrt{x}\) represents a value that, when multiplied by itself, gives \(x\).
• For any positive value of \(x\), \(\sqrt{x}\) is real. However, square roots of negative numbers are not real and fall into the category of complex numbers.
• In square root functions, as \(x\) becomes larger, \(\sqrt{x}\) increases, but at a decreasing rate.

In inverse variation, we are often interested in how another variable, like \(y\), changes in relation to \(\sqrt{x}\). Understanding these functions allows us to interpret and solve equations involving square roots.

Algebraic Equations

Algebraic equations are mathematical rules that use symbols and letters to represent numbers and operations. In the inverse variation example, we set up an algebraic equation to describe how \(y\) varies inversely with the square root of \(x\).

• An algebraic equation like \(y = \frac{k}{\sqrt{x}}\) indicates a specific relationship between the variables.
• The symbols represent the quantities being related - \(y\) and \(x\) in this case.
• Equations can usually be solved to find the values of unknowns given certain conditions.

We often need to determine constants like \(k\) by substituting known values into the equation, making it a practical tool for relationships between quantities.

Constant of Variation

A constant of variation serves as a bridge in equations where variables are connected by a specific rule. In our exercise, the constant of variation \(k\) is found to maintain the relationship between \(y\) and \(\sqrt{x}\) consistent.

• The constant \(k\) in \(y = \frac{k}{\sqrt{x}}\) ensures that the equation holds true for all applicable \(x\) and \(y\) values.
• Determining \(k\) often involves solving the equation with known variable values, as done in the provided step-by-step solution.
• Once \(k\) is known, you can predict how changes in \(x\) will affect \(y\) and vice versa.

The constant of variation is essential for keeping proportional relationships stable, and understanding it is key to mastering inverse variation problems.