Problem 6 - Algebra and Trigonometry with Analytic Geometry

Question

Express the statement as a formula that involves the given variables and a constant of proportionality \(k\), and then determine the value of \(k\) from the given conditions. \(q\) is inversely proportional to the sum of \(x\) and \(y\). If \(x=0.5\) and \(y=0.7\), then \(q=1.4\).

Step-by-Step Solution

Answer

The constant of proportionality \(k\) is 1.68.

Step 1: Understand Inverse Proportionality

When a variable \(q\) is inversely proportional to another variable (or expression), it means \(q\) is equal to a constant divided by that variable or expression. Here, \(q\) is inversely proportional to the sum of \(x\) and \(y\), meaning: \[ q = \frac{k}{x + y} \] where \(k\) is the constant of proportionality.

Step 2: Write Down the Formula

Based on the inverse proportionality, express \(q\) using the equation: \[ q = \frac{k}{x+y} \] This sets the mathematical relationship between \(q\), \(x\), \(y\), and \(k\).

Step 3: Substitute Given Values

To find the constant \(k\), substitute the given values into the equation. We have \(q = 1.4\), \(x = 0.5\), and \(y = 0.7\). Substitute these into the equation: \[ 1.4 = \frac{k}{0.5 + 0.7} \]

Step 4: Solve for k

First, calculate the sum of \(x\) and \(y\): \[ 0.5 + 0.7 = 1.2 \]Next, substitute this into the equation to find \(k\): \[ 1.4 = \frac{k}{1.2} \]Multiply both sides by 1.2 to solve for \(k\): \[ k = 1.4 \times 1.2 = 1.68 \]

Key Concepts

Constant of Proportionality

The constant of proportionality, denoted as \(k\), is a vital concept when dealing with proportional relationships. It represents the fixed ratio in these relationships, dictating how one variable changes in response to another.

In the realm of inverse proportionality, the constant \(k\) inversely relates a variable to another. This means that one variable increases as the other decreases, maintaining the product equal to \(k\).

In our example, \(q\) is inversely proportional to the sum of \(x\) and \(y\), indicating that \(q\) becomes smaller as \(x + y\) becomes larger, following the equation:

• \( q = \frac{k}{x+y}\)

To find \(k\), you substitute the given values and solve the equation. Here, substituting \(x = 0.5\), \(y = 0.7\), and \(q = 1.4\) helps calculate the value of \(k\), which is 1.68.

Mathematical Relationship

A mathematical relationship provides a structured way to understand how different quantities are connected.

In cases of proportionalities, such as direct or inverse, we use algebraic formulas to express these relationships.

For inverse proportionality, like in this exercise, the variable \(q\) inversely relates to another expression, specifically \(x + y\). The connection is established by the formula:

• \( q = \frac{k}{x+y} \)

This equation tells us how \(q\) will change depending on \(x\) and \(y\).

Through these formulas, we can determine the values of unknown variables once specific conditions are provided, reaffirming the relevance of understanding these mathematical relationships.

Algebraic Expressions

Algebraic expressions serve as a fundamental part of forming mathematical models. They comprise variables, constants, and arithmetic operations, allowing us to convey mathematical ideas succinctly.

In the context of inverse proportionality, an algebraic expression helps represent the relationship between various quantities. Here, the expression \(x + y\) contributes to forming the denominator in the inverse relationship with \(q\):

• \( q = \frac{k}{x+y} \)

Such expressions enable us to convert verbal statements of mathematical relationships into concrete formulas.

These expressions thus act as the language of mathematics, empowering us to solve complex problems with ease and clarity.