Problem 17 - Algebra and Trigonometry

Question

13–22 ? Express the statement as an equation. Use the given information to find the constant of proportionality. \(W\) is inversely proportional to the square of \(r .\) If \(r=6,\) then \(W=10 .\)

Step-by-Step Solution

Answer

The constant of proportionality is 360.

Step 1: Understand Inverse Proportionality

When a quantity is inversely proportional to another, it means that as one quantity increases, the other decreases in such a way that their product remains constant. Here, we are told that \( W \) is inversely proportional to the square of \( r \). This implies that \( W \times r^2 = k \), where \( k \) is the constant of proportionality.

Step 2: Set Up the Equation

Express the inverse proportionality as an equation. Since \( W \) is inversely proportional to \( r^2 \), we can write the equation as:\[ W = \frac{k}{r^2} \] where \( k \) is the constant of proportionality we need to find.

Step 3: Substitute Known Values

We are given that \( r = 6 \) and \( W = 10 \). Substitute these values into the equation to find \( k \):\[ 10 = \frac{k}{6^2} \]

Step 4: Solve for the Constant of Proportionality

Since \( 6^2 = 36 \), our equation becomes:\[ 10 = \frac{k}{36} \]Multiply both sides by 36 to solve for \( k \):\[ k = 10 \times 36 = 360 \]

Step 5: Verify the Equation

Now that we have \( k = 360 \), substitute back into the inverse proportionality equation to verify:\[ W = \frac{360}{r^2} \] For \( r = 6 \), \( W = \frac{360}{36} = 10 \), which is consistent with the given values.

Key Concepts

Understanding the Constant of Proportionality

The constant of proportionality is a fixed value that relates two variables in a proportional relationship. In our problem, this constant is denoted by \( k \). When quantities are related by inverse proportionality, the constant ensures that as one variable increases, the other decreases in such a way that their product stays the same.

Here's a simple breakdown:

• In a typical inverse proportionality scenario, the equation takes the form \( x \times y = k \).
• In our specific example, we are given \( W \times r^2 = k \).
• This equation tells us how \( W \) and \( r^2 \) relate to each other through this constant \( k \), that keeps their product unchanged.

Understanding this constant helps us see how two quantities can be connected yet move in opposite directions when one is increased or decreased.

Grasping Inverse Variation

Inverse variation describes a relationship between two variables where one variable increases as the other decreases, in such a way that their product remains constant. This connection is the essence of inverse proportionality.

With inverse variation:

• The equation we use is \( y = \frac{k}{x} \), where \( k \) is the constant of proportionality.
• In our problem, the relationship of \( W \) with \( r^2 \) is described as \( W = \frac{k}{r^2} \).
• As \( r \) increases, \( r^2 \) increases as well, causing \( W \) to decrease for the equation to balance out.

This inverse connection means the two quantities move in opposite directions, while their product remains fixed at \( k \). This type of variation is commonly found in many real-world situations where one factor compensates for changes in another.

Steps to Equation Solving

Solving equations with inverse proportionality involves understanding the setup and carefully replacing values to find unknowns. In our problem:

Here's the process step-by-step:

• Write the inverse proportionality equation based on the given relationship, in this case, \( W = \frac{k}{r^2} \).
• Substitute values that you know. Here, \( r = 6 \) and \( W = 10 \).
• Calculate to find \( k \): Substitute into the equation and solve for \( k \), giving us \( 10 = \frac{k}{36} \).
• Multiply both sides by the denominator to find \( k \): Here, \( k = 360 \).
• VERIFY: Substitute \( k \) back into your equation to check the correctness. Ensure your solution satisfies the original condition.

This process of substituting known values, manipulating the equation, and verifying the result, is fundamental in equation solving. It assures that the values make sense and adhere to the original problem statement.