Problem 8 - Introductory Algebra for College Students

Question

Determine the constant of variation for each stated condition. \(T\) varies inversely as \(n,\) and \(T=4\) when \(n=24\)

Step-by-Step Solution

Answer

The constant of variation is 96.

Step 1: Understanding Inverse Variation

Let's first clarify the concept of inverse variation. This means that one variable increases as the other decreases or vice versa. In this exercise, \(T\) varies inversely as \(n\). The mathematical representation for this relationship is \(T = k/n\) where \(k\) is the constant of variation.

Step 2: Substitute the given values into the equation

We are told that \(T=4\) when \(n=24\). So, we substitute these values into our equation, \(T = k/n\), to get \(4 = k/24\).

Step 3: Solve for k

To find \(k\), we now need to solve the equation \(4 = k/24\). We do this by multiplying both sides of the equation by 24. This simplifies to \(k = 4*24 = 96\).

Key Concepts

Understanding Inverse Variation

Inverse variation describes a specific mathematical relationship where one variable increases as the other decreases, and vice versa. When we discuss the notion of inverse variation in algebra, we're often referring to the foundational expression \( y = \frac{k}{x} \) where \( k \) is known as the constant of variation.

In the given exercise, \( T \) varies inversely as \( n \) which means that as \( n \) grows, \( T \) becomes smaller according to their inverse relationship. Conversely, as \( n \) decreases, \( T \) increases. The formula capturing this relationship is \( T = \frac{k}{n} \), with \( k \) representing the constant of variation to be determined.

Algebraic Equations and Inverse Variations

Algebraic equations are statements of equality that feature variables, numbers, and sometimes exponents or roots. They are fundamental in expressing relationships between quantities and finding unknown values.

In the context of inverse variation, the algebraic equation takes a specific form, \( y = \frac{k}{x} \), where \( k \) is a non-zero constant, and \( x \) and \( y \) are the variables inversely related to each other. To solve such equations, we perform algebraic operations to isolate the variable or constant we want to find. For instance, if we are given values for \( y \) and \( x \) and we need to find \( k \) as in our exercise, we would multiply both sides by \( x \) to solve for \( k \) thereby deriving the constant of variation.

Solving for Variables

Solving for variables is an essential part of algebra, often requiring manipulation of equations to isolate the desired variable. When we're given an algebraic equation with a variable hidden within it, our goal is to perform operations that will 'free' this variable, making it the subject of the formula.

In our exercise, solving for the constant of variation \( k \) falls under this process. We started with substituting known variable values into the inverse variation equation, setting the stage for finding \( k \). Multiplication is used to eliminate the fraction. By multiplying both sides by the variable \( n \) and simplifying, we isolated \( k \) to obtain its value, which turns out to be \( 96 \) given that \( T=4 \) and \( n=24 \) in the initial conditions.