Problem 23 - Beginning and Intermediate Algebra

Question

Write a general variation equation using \(k\) as the constant of variation. Suppose \(Q\) varies directly as the square of \(r\) and inversely as \(w .\) If \(Q=25\) when \(r=10\) and \(w=20\) a) find the constant of variation. b) write the specific variation equation relating \(Q, r\) and \(w\) c) find \(Q\) when \(r=6\) and \(w=4\)

Step-by-Step Solution

Answer

The constant of variation is \(k = 5\), and the specific variation equation is \(Q = 5 \cdot \frac{r^2}{w}\). When \(r = 6\) and \(w = 4\), the value of \(Q\) is 45.

Step 1: Write the general variation equation

Since \(Q\) varies directly with the square of \(r\) and inversely with \(w\), we can write the general variation equation as: \[Q = k \cdot \frac{r^2}{w}\] Here, \(k\) is the constant of variation that we need to determine.

Step 2: Find the constant of variation \(k\)

We are given that \(Q = 25\), \(r = 10\), and \(w = 20\). We can plug these values into our general variation equation to find \(k\): \[25 = k \cdot \frac{10^2}{20}\] Now, let's solve the equation for \(k\): \[k = \frac{25 \cdot 20}{10^2}\] \[k = \frac{500}{100}\] \[k = 5\] So, the constant of variation is \(k = 5\).

Step 3: Write the specific variation equation

Now that we have the constant of variation, let's plug it into our general variation equation to write the specific variation equation: \[Q = 5 \cdot \frac{r^2}{w}\] This is the specific variation equation relating \(Q\), \(r\), and \(w\).

Step 4: Find the value of \(Q\) when \(r=6\) and \(w=4\)

We are asked to find the value of \(Q\) when \(r = 6\) and \(w = 4\). We can plug these values into our specific variation equation to find \(Q\): \[Q = 5 \cdot \frac{6^2}{4}\] \[Q = 5 \cdot \frac{36}{4}\] \[Q = 5 \cdot 9\] \[Q = 45\] So, when \(r = 6\) and \(w = 4\), the value of \(Q\) is 45.

Key Concepts

Direct variation

In the context of variation equations, direct variation signifies that one variable increases (or decreases) in proportion to the powers of another variable. When something varies directly, it means that as one variable goes up, the other does so proportionally, assuming the direct proportionality constant remains unchanged. For our exercise, this is clearly demonstrated by the way variable \(Q\) changes in relation to the square of \(r\). This relationship can be expressed using a basic equation:

• If two quantities, \(y\) and \(x\), are directly proportional and\(x^2\), then \(y=kx^2\), where \(k\) is the constant of variation.

In our specific exercise, this means the bigger \(r\) gets, \(Q\) will also get bigger because \(Q\) increases with the square of \(r\). This pattern of change is what defines the direct variation feature present in the equation.

Inverse variation

Inverse variation describes a situation where one variable increases while another variable decreases, and their product remains constant. This relationship appears whenever a variable is inversely proportional to another, meaning that as one variable becomes larger, the other becomes smaller, given the inverse proportionality constant does not change.

• If \(y\) is inversely proportional to \(x\), the relationship is expressed as \(y = \frac{k}{x}\), where \(k\) is the constant of variation.

In the exercise, the variable \(Q\) is inversely proportional to \(w\). Hence, if \(w\) increases, \(Q\) decreases, showcasing the classic inverse relationship. This keeps the product of \(Q\) and \(w\) consistent in proportion due to the presence of \(k\). Understanding these relationships aids in solving problems that involve calculating variations dependent on multiple conditions.

Constant of Variation

The constant of variation, represented as \(k\) in variation equations, serves as a proportionality factor that stabilizes the equation, dictating the level at which variables relate to each other. It can be thought of as the glue of a variation equation, ensuring that the computed values remain consistent with empirical data or defined conditions.

• The constant of variation \(k\) remains constant for all calculations for given conditions.

To find \(k\), specific values of the other variables in the equation are substituted, as seen in the step-by-step solution. In our example, knowing specific values of \(Q\), \(r\), and \(w\), allowed us to solve for \(k=5\). This constant helps translate the theoretical variation equation into a practical tool, tying theoretical mathematics to real-world values through normalizing the equation.

General variation equation

A general variation equation is a mathematical representation that captures the collective essence of various types of variations, commonly combining aspects of both direct and inverse variations. This comprehensive type of equation shows how different variables interact under a specified rule of variation.

• In the exercise, the general form is \(Q = k \cdot \frac{r^2}{w}\).

This equation shows that \(Q\) is proportional to the square of \(r\) (direct variation) and inversely proportional to \(w\). The presence of both direct and inverse components combines their distinct characteristics into one unified equation, offering a detailed map of how the variables coalesce under the given mathematical constraints. Such types of equations are essential in describing complex systems where multiple factors exert different kinds of influences on a single outcome.