Problem 21 - Precalculus Mathematics for Calculus

Question

Express the statement as an equation. Use the given information to find the constant of proportionality. \(M\) varies directly as \(x\) and inversely as \(y .\) If \(x=2\) and \(y=6\) then \(M=5\).

Step-by-Step Solution

Answer

The constant of proportionality, \(k\), is 15.

Step 1: Define Proportionality Relationship

The statement that \(M\) varies directly as \(x\) and inversely as \(y\) can be expressed as an equation: \(M = k \frac{x}{y}\), where \(k\) is the constant of proportionality.

Step 2: Substitute Known Values

Substitute the known values into the equation: \(M = 5\), \(x = 2\), and \(y = 6\). This gives us the equation \(5 = k \frac{2}{6}\).

Step 3: Solve for Constant of Proportionality

Rearrange the equation to solve for \(k\). First, simplify \(\frac{2}{6}\) to \(\frac{1}{3}\), giving the equation \(5 = k \frac{1}{3}\). Multiply both sides by 3 to isolate \(k\): \(k = 15\).

Key Concepts

Constant of Proportionality

In mathematics, when quantities vary directly or inversely with each other, they are said to be either directly or inversely proportional. A crucial part of solving such problems is finding the "constant of proportionality," denoted as \(k\). This constant helps describe how two variables relate to each other. For a situation where a quantity \(M\) varies directly as \(x\) and inversely as \(y\), the relationship can be represented by the formula: \[ M = k \frac{x}{y} \] In this setup:

• The variable \(x\) is directly proportional to \(M\), meaning as \(x\) increases, so does \(M\) if \(y\) remains constant.
• The variable \(y\) is inversely proportional to \(M\), indicating that \(M\) decreases as \(y\) increases, given \(x\) is constant.

To find the constant \(k\), use known values of \(M\), \(x\), and \(y\). Once \(k\) is determined, it helps establish the exact relationship between these quantities.

Proportionality Equation

A proportionality equation captures the relationship between variables where one variable is proportional to another. In our context, if we say \(M\) varies directly as \(x\) and inversely as \(y\), we describe this with the proportionality equation: \[ M = k \frac{x}{y} \] Here, this equation combines both direct and inverse relationships. It tells us:

• If you double \(x\), then \(M\) will also double, assuming \(y\) remains unchanged.
• Conversely, if you double \(y\), \(M\) will halve, as long as \(x\) is constant.

This type of equation allows us to predict one variable if the other two are known, reinforcing the interdependent nature of these variables.

Solving Equations

Solving an equation involving direct and inverse variation is about rearranging the equation to find unknown quantities. For example, to find the constant of proportionality \(k\), you need to plug in given values and solve. Consider the equation: \[ M = k \frac{x}{y} \] Given \(M = 5\), \(x = 2\), and \(y = 6\), you substitute these values: \[ 5 = k \frac{2}{6} \] By simplifying \(\frac{2}{6}\) to \(\frac{1}{3}\), you get: \[ 5 = k \times \frac{1}{3} \] Then, multiply both sides by 3 to solve for \(k\): \[ k = 15 \] This solution tells us the constant \(k\) that links \(M\), \(x\), and \(y\). Through this process, you're translating a problem in words into mathematical expressions to derive exact values.