Problem 14 - College Algebra and Calculus: An Applied Approach

Question

Write a linear model that relates the variables. \(r\) varies directly as \(s ; r=25\) when \(s=40\)

Step-by-Step Solution

Answer

The linear model that relates the variables \(r\) and \(s\) is \(r = 0.625s\).

Step 1: Calculate the constant of variation

First, we need to find the constant of variation in a direct variation. We can do this by using the specific conditions given, where \(r = 25\) when \(s = 40\). We can substitute these values into our general equation \(r = ks\) to get \(25 = k \cdot 40\). Solving this equation for \(k\), we find \(k = 25/40 = 0.625\).

Step 2: Write the linear model

Now that we have determined the constant of variation, we can write the final linear model that relates \(r\) and \(s\). Replacing \(k\) in our general equation \(r = ks\), we get \(r = 0.625s\). This is the equation that expresses the direct variation between \(r\) and \(s\).

Key Concepts

Linear Model

A linear model is a straightforward mathematical relationship between two variables. In many practical situations, one variable depends directly on another, and this is where a linear model comes into play. Imagine you have two variables, say \(r\) and \(s\), and you know that \(r\) varies directly as \(s\). This means wherever there's a change in \(s\), \(r\) changes in a consistent way. The formula for this relationship can be written as \(r = ks\), where \(k\) is the constant of variation. This equation is a linear model because it graphically depicts a straight line when plotted, with \(r\) on one axis and \(s\) on the other. Linear models are crucial in predicting outcomes and understanding relationships between data points. They simplify complex relationships into a linear form that is easy to work with and analyze.

• Straight-line graphical representation.
• Expresses proportional relationships.
• Used in making predictions based on existing data.

Constant of Variation

The constant of variation, denoted as \(k\), is the key factor in a direct variation scenario. It defines the rate at which two variables are related linearly. To find \(k\), you take the known values of variables \(r\) and \(s\) and substitute them into the formula \(r = ks\). For the exercise you have, when \(r = 25\) and \(s = 40\), plugging these into the formula gives you \(25 = k \cdot 40\). To find \(k\), solve the equation by dividing both sides by 40, yielding \(k = 0.625\).
This constant essentially indicates how much \(r\) changes for a one-unit change in \(s\). It is a critical value that shapes the slope of the line in a linear model, providing insight into the relationship's intensity.

• Determines the slope of the linear equation.
• Essential for understanding proportional relationships.
• Calculated by dividing one variable by the other.

Algebra

Algebra is a branch of mathematics dealing with symbols and rules for manipulating those symbols. It is essential when working with direct variation and linear models, as it provides the tools to derive formulas and solve equations. In the given problem, algebraic techniques are used to find the constant of variation and to express the relationship between the variables in a mathematical form.
To solve for the constant of variation, the algebraic manipulation involves substituting known values into the equation \(r = ks\), and solving for the unknown quantity \(k\) by isolating it on one side of the equation. By understanding how to perform these operations, algebra allows you to express complex mathematical relationships in simpler terms.

• Involves solving equations and manipulating variables.
• Foundation for understanding direct variation and relationships.
• Empowers you to transform real-world situations into mathematical expressions.