Problem 13 - College Algebra and Calculus: An Applied Approach

Question

Write a linear model that relates the variables. \(c\) is proportional to \(d ; c=12\) when \(d=20\)

Step-by-Step Solution

Answer

The linear model relating \(c\) and \(d\) is \(c = 0.6 * d\).

Step 1: Formulate the Proportional Relationship

Initially, the proportional relationship between \(c\) and \(d\) is given by the equation \(c = k * d\). Here, \(k\) is the constant of proportionality.

Step 2: Substitute Known Values

Next, substitute the values \(c = 12\) and \(d = 20\) into the equation. This gives: \(12 = k * 20\).

Step 3: Solve for the Constant of Proportionality

Finally, solve the equation for \(k\) to find the constant of proportionality. This gives \(k = 12 / 20\) or \(k = 0.6\) after simplifying.

Step 4: Write the Final Model

Now that we have the constant of proportionality, we can write the final linear model. It is given by: \(c = 0.6 * d\).

Key Concepts

Proportional Relationship

Understanding proportional relationships is fundamental when working with linear models. In simple terms, a proportional relationship between two variables implies that as one variable increases or decreases, the other does so at a consistent rate. This consistent rate is known as the constant of proportionality.

For example, consider you're running a lemonade stand and the cost of lemons is directly proportional to the weight of lemons you buy. If the cost doubles when you buy double the weight, there is a proportional relationship. Mathematically, this is expressed as one variable being equal to the other variable multiplied by a constant. If we let cost be represented by the variable 'c', and weight by 'd', our proportional relationship can be described by the equation \( c = kd \), where 'k' is our constant of proportionality.

Constant of Proportionality

The constant of proportionality, symbolized as \( k \) in equations, is what determines the rate at which the proportional relationship affects the variables in question. It is, in essence, the 'fixed ratio' in a proportional relationship.

Using our lemonade stand example, if the cost (\( c \)) of lemons for 20 kilograms (\( d \)) is \(12, we can find the constant of proportionality by setting up the equation \( 12 = k \times 20 \). Simplifying, \( k = \frac{12}{20} = 0.6 \), stating that for every kilogram of lemons, the cost is \)0.6. The significance of finding this constant is immense, as it allows us to predict the cost for any weight of lemons, making it a powerful tool for planning and budgeting.

Solving Equations

When we have an equation like the one in our lemonade stand scenario, solving for the unknown, in this case 'k', requires an understanding of basic algebraic principles. Solving equations often involves isolating the variable of interest on one side of the equation using operations such as addition, subtraction, multiplication, or division.

In our example, we got the equation \( 12 = k \times 20 \). To isolate \( k \), we divide both sides of the equation by 20, giving us \( k = \frac{12}{20} \), which we then simplify to \( k = 0.6 \). This process of solving for the constant of proportionality in proportional relationships is a linchpin in creating linear models, which are ubiquitous in various fields such as economics, science, and engineering.