Problem 10 - Precalculus: Functions and Graphs

Question

Exer. 1-12: Express the statement as a formula that involves the given variables and a constant of proportionallty \(k,\) and then determine the value of \(k\) from the given conditions. \(y\) is directly proportional to \(x\) and inversely proportional to the sum of \(r\) and \(s .\) If \(x=3, r=5,\) and \(s=7,\) then \(y=2\)

Step-by-Step Solution

Answer

The constant of proportionality \(k\) is 8.

Step 1: Understand the Proportionality

The statement 'y is directly proportional to x' means that as \(x\) increases, \(y\) also increases, and 'y is inversely proportional to the sum of \(r\) and \(s\)' means as the sum \(r+s\) increases, \(y\) decreases. This creates a proportional relationship which can be modeled as \(y = k \frac{x}{r+s}\), where \(k\) is the constant of proportionality.

Step 2: Formulate the Equation

Given the above understanding, we can write the equation in terms of the variables and the constant of proportionality: \[ y = k \frac{x}{r+s} \]

Step 3: Substitute Given Values into the Equation

Using the provided values, substitute \(x = 3\), \(r = 5\), \(s = 7\), and \(y = 2\) into the proportional relationship equation to find \(k\): \[ 2 = k \frac{3}{5+7} \]

Step 4: Simplify and Solve for k

First, simplify the denominator: \(5+7 = 12\). Substitute into the equation: \[ 2 = k \frac{3}{12} \] \[ 2 = k \frac{1}{4} \] To isolate \(k\), multiply both sides by 4: \[ k = 8 \]

Key Concepts

Constant of Proportionality

When we talk about proportional relationships, the constant of proportionality often plays a crucial role. It is a fixed number that relates the quantities in a proportional relationship. In our given problem, the proportional relationship is formed between two expressions: one directly and one inversely. This is expressed mathematically using the formula:

• For direct proportionality, if "y is directly proportional to x," as x increases, y increases too. The constant makes this relationship precise by acting as a multiplier.
• For inverse proportionality, "y is inversely proportional to the sum of r and s," meaning that as the sum changes, the effect on y is the opposite. The constant relates this inverse change.

Here, the equation that ties all parts of the problem together is \[ y = k \frac{x}{r+s} \]The constant "k" tells us how much y will change given changes in x, r, and s. In this example, once we substitute the given values, the constant of proportionality k turned out to be 8, after solving the equation based on these relationships.

Algebraic Equations

An algebraic equation is a mathematical statement that uses variables to show relations between different quantities. In our problem, an equation is essential for finding the relationship between y, x, r, and s.
The equation given is:

• \(y = k \frac{x}{r+s}\)

This equation comprises variables y, x, r, and s, as well as the constant k. To use this equation effectively:

• Understand that solving it involves substitution of known values into the equation.
• Rearrange or solve for the unknown, here "k," which in this case was done by isolating k after substituting the provided values.

The result is an equation that not only describes the relationship but also allows you to find that missing constant. Equations like these are prevalent in math and science because they give us a way to express complex relationships simply and solve for unknowns with straightforward algebraic manipulation.

Proportional Relationships

Proportional relationships indicate how two quantities relate to each other. There are typically two kinds: direct and inverse proportionality. Let’s break it down in our context:

• A direct proportional relationship means that when one variable increases, the other also increases. Here, y is directly proportional to x, so as x becomes larger, y will also grow, assuming other variables remain constant.
• An inverse proportional relationship means that when one variable increases, the other decreases. In this equation, y is inversely proportional to the sum \(r+s\). This means if the sum of r and s increases, y will decrease, again assuming that x remains the same.

Such relationships can be seen in the equation: \[ y = k \frac{x}{r+s} \]Recognizing the type of proportional relationship helps you determine how changes in one factor affect another. By understanding these relationships, even complex systems can be simplified into more manageable mathematical expressions or algebraic equations. This makes solving practical problems more efficient and comprehensible.