Problem 7 - Introductory Algebra for College Students

Question

Determine the constant of variation for each stated condition. \(W\) varies inversely as \(r,\) and \(W=600\) when \(r=10\)

Step-by-Step Solution

Answer

The constant of variation \(k\) is 6000.

Step 1: Understand the equation

The equation for inverse variation is \(W = k/r\) where \(W\) is the variable that represents the dependent quantity, \(r\) is the variable that represents the independent quantity, and \(k\) is the constant of variation.

Step 2: Insert known values into the equation

Given, \(W = 600\) when \(r = 10\). Substitute these values into the equation: \(600 = k/10\).

Step 3: Solve for k

To solve for \(k\), multiply both sides of the equation by 10: \(k = 600 * 10 = 6000\).

Key Concepts

Inverse Variation

In an inverse variation situation, two quantities are related in such a way that as one quantity increases, the other decreases at an equal rate, and vice versa. This relationship can be described by the equation \( W = \frac{k}{r} \) where \( W \) is the dependent variable, \( r \) is the independent variable, and \( k \) is the constant of variation.

In practical terms, this means that if \( r \) is doubled, for example, \( W \) would be halved, assuming \( k \) remains unchanged. This concept is essential in many fields, including physics, where it describes phenomena like the intensity of light inversely varying with the square of the distance from the light source.

To determine the constant of variation, one must have the values for both \( W \) and \( r \) for a specific situation. Once these values are known, they can be substituted into the equation to solve for \( k \) as demonstrated in the original exercise.

Algebraic Equations

Algebraic equations are mathematical statements that show the relationship between different variables. They consist of expressions on both sides of an equal sign. In the context of inverse variation, the equation takes the form \( W = \frac{k}{r} \) which shows that \( W \) is proportional to the inverse of \( r \) with \( k \) being the proportionality constant.

In algebra, we often manipulate equations to solve for a particular variable. This involves performing the same operation on both sides of the equation to maintain equality, as was seen in solving for the constant of variation. Algebraic equations can be simple or complex and can have one or more variables. Understanding how to manipulate these equations is fundamental to solving mathematical problems across numerous disciplines.

Solving for Variables

Solving for variables is a fundamental skill in algebra that allows one to find the value of an unknown quantity. This process involves isolating the variable on one side of the equation. In our example with inverse variation, solving for the constant of variation, \( k \) involves manipulating the equation \( W = \frac{k}{r} \) such that \( k \) is on one side by itself.

To isolate \( k \) in our example, we multiplied both sides of the equation by \( r \), which gave us \( k = W \cdot r \) since multiplying by \( r \) cancels out the division by \( r \) on the right side of the equation. Such manipulation not only helps in finding the value of \( k \) but is also a technique used across various types of algebraic equations to find the values of different unknowns.