Problem 104 - College Algebra with Modeling and Visualization

Question

Let \(y\) be directly proportional to \(x\) Complete the following. Find \(y\) when \(x=1.3,\) if \(y=7.2\) when \(x=5.2\)

Step-by-Step Solution

Answer

When \( x = 1.3 \), \( y \approx 1.8 \).

Step 1: Understand Direct Proportion

When we say that one quantity is directly proportional to another, it means they increase or decrease together. In mathematical terms, if \( y \) is directly proportional to \( x \), then \( y = kx \) where \( k \) is the constant of proportionality.

Step 2: Find the Constant of Proportionality

Given that \( y = 7.2 \) when \( x = 5.2 \), we can use these values to find \( k \). Substitute them into the equation \( y = kx \) to get: \[7.2 = k imes 5.2\]Solve for \( k \) by dividing both sides by 5.2:\[k = \frac{7.2}{5.2} = 1.384615\]

Step 3: Use the Constant to Find y for a New x

Now that we have \( k = 1.384615 \), use it to find \( y \) when \( x = 1.3 \). Substitute \( x = 1.3 \) and \( k = 1.384615 \) into the equation \( y = kx \):\[y = 1.384615 \times 1.3\]Calculate the expression:\[y = 1.7999995 \approx 1.8\]

Step 4: Round to the Appropriate Decimal Place

The calculation gives \( y \approx 1.8 \). Since the values given have one decimal place, it is appropriate to present the answer as \( 1.8 \).

Key Concepts

Constant of Proportionality

The constant of proportionality, often denoted as \( k \), is a fundamental concept in understanding direct proportion. It serves as the link between two quantities that scale together. If \( y \) is directly proportional to \( x \), the relationship can be expressed with the equation \( y = kx \). Here, \( k \) indicates how much \( y \) changes with a given change in \( x \).

For instance, to find \( k \) when given that \( y = 7.2 \) and \( x = 5.2 \), plug these values into the equation: \[ 7.2 = k imes 5.2. \] Solve for \( k \) by dividing by \( 5.2 \): \[ k = \frac{7.2}{5.2} \approx 1.384615. \]

• \( k \) remains constant for all values of \( x \) and \( y \) in a proportional relationship.
• It shows the rate at which \( y \) increases as \( x \) increases, and vice versa.

Algebraic Equations

Algebraic equations are crucial when dealing with direct proportion problems. They provide a precise mathematical framework to establish relationships between variables. When \( y \) is directly proportional to \( x \), the equation \( y = kx \) describes this relationship.

This equation signifies that for every unit increase in \( x \), \( y \) increases by \( k \) units. To solve these equations, identifying the constant of proportionality \( k \) is essential. For example, with known values of \( y = 7.2 \) and \( x = 5.2 \), the equation becomes \( 7.2 = k imes 5.2 \). Solving for \( k \) gives \( k \approx 1.384615 \).

• The algebraic equation \( y = kx \) is a linear equation, forming a straight line when graphed.
• Once \( k \) is determined, you can easily find \( y \) for any other \( x \) by substituting \( x \) into the equation \( y = kx \).

Mathematical Modeling

Mathematical modeling is a powerful tool for representing real-world scenarios with mathematical equations. Direct proportionality is one such model that depicts consistent relationships between quantities. When using the equation \( y = kx \), you are effectively modeling how \( y \)'s value changes in direct response to changes in \( x \).

Take the given condition: "\( y = 7.2 \) when \( x = 5.2 \)." Using this to calculate the constant \( k \), permits us to forecast how \( y \) will behave should \( x \) change. For example, if \( x = 1.3 \), substituting into the model \( y = 1.384615 \times 1.3 \) gives \( y \approx 1.8 \).

• Math modeling simplifies complex relations into standard formulas.
• It allows predictions of unknown values, like finding \( y \) for an unspecified \( x \).
• This approach is widely used in sciences to predict outcomes and understand proportional changes.