Problem 61 - Thomas Calculus

Question

The variables \(r\) and \(s\) are inversely proportional, and \(r=6\) when \(s=4 .\) Determine \(s\) when \(r=10 .\)

Step-by-Step Solution

Answer

When \( r = 10 \), \( s \approx 2.4 \).

Step 1: Identify Relationship

Since the variables are inversely proportional, we use the formula: \( r \times s = k \), where \( k \) is a constant.

Step 2: Find the Constant of Proportionality

Use the given values \( r = 6 \) and \( s = 4 \) to find \( k \). Substitute these into the equation: \( 6 \times 4 = k \). Thus, \( k = 24 \).

Step 3: Set Up the Equation with New Value of r

Substitute \( r = 10 \) and the constant \( k = 24 \) back into the inverse proportionality equation: \( 10 \times s = 24 \).

Step 4: Solve for s

Divide both sides of the equation \( 10 \times s = 24 \) by 10 to solve for \( s \): \( s = \frac{24}{10} = 2.4 \).

Key Concepts

Constant of Proportionality

When dealing with inverse proportionality, it's essential to understand the concept of a constant of proportionality. For two variables to be inversely proportional, their product must always equal the same constant. This is expressed mathematically as \( r \times s = k \), where \( k \) is the constant of proportionality.

In the given problem, we started with values \( r = 6 \) and \( s = 4 \). By multiplying these together, we found that the constant \( k \) is 24, because \( 6 \times 4 = 24 \).

Once you know \( k \), it becomes a powerful tool to resolve similar future problems. It remains unchanged for all corresponding values of the variables, no matter how they change, as long as the inverse relationship holds.

Relationship between Variables

Understanding the relationship between variables is crucial when tackling problems involving inverse proportionality.

In an inverse proportional relationship, as one variable increases, the other decreases proportionally, and vice versa. This means that the product of these two variables will always remain constant. This is expressed in the equation \( r \times s = k \).

In our context, as the value of \( r \) increases from 6 to 10, the value of \( s \) necessarily decreases to maintain the constant product of 24. This highlights the characteristic relationship between inversely proportional variables.

Solving Equations

Solving equations involving inverse proportional relationships can be straightforward using the constant of proportionality.

Given the relationship \( r \times s = 24 \), if one variable changes, you can solve for the other by rearranging the equation accordingly. In the problem, to find \( s \) when \( r = 10 \), we substitute into the equation to get \( 10 \times s = 24 \).

To find the value of \( s \), divide both sides by 10, resulting in \( s = \frac{24}{10} = 2.4 \). This process demonstrates how rearranging and solving equations allows us to find unknown values efficiently while maintaining the integrity of the inverse relationship.