Problem 13 - Introductory Algebra for College Students

Question

\- A person's fingernail growth, \(G\), in inches, varies directly ?s the number of weeks it has been growing, \(W\). Write an equation that expresses this relationship. Fingernails grow at a rate of about 0.02 inch per week. Substitute 0.02 for \(k,\) the constant of variation, in the equation in part (a) and write the equation for fingernail growth. =c. Substitute 52 for \(W\) to determine your fingernail growth at the end of one year if for some bizarre reason you decided not to cut them and they did not break.

Step-by-Step Solution

Answer

If you did not cut your fingernails and they did not break for a year, they would grow approximately \(1.04\) inches.

Step 1: Establish the Relationship

Given that the growth \(G\) varies directly with the number of weeks \(W\), this relationship can be expressed in general form: \(G = kW\) where \(k\) is the constant of variation.

Step 2: Substitute the Constant of Variation

It is given that finger nails grow at a rate of about 0.02 inch per week, so this is the value for \(k\). Substitute this into the equation: \(G = 0.02W\). This is the equation for fingernail growth.

Step 3: Determine Fingernail Growth for One Year

Substitute 52 for \(W\) in the equation \(G = 0.02W\) as there are 52 weeks in a year: \(G = 0.02 * 52\). Solve this equation to find fingernail growth at the end of one year.

Key Concepts

Algebraic Relationships

In algebra, we come across various kinds of relationships between sets of numbers or variables. A direct variation is one such fundamental relationship where two variables change in proportion to each other. For instance, in our exercise, we talk about a person's fingernail growth (\(G\)) varying directly as the number of weeks (\(W\)) it has been growing. This implies that as the number of weeks increases, the fingernail length also increases in a predictable way. The general formula for direct variation is \(Y = kX\), where \(Y\) and \(X\) are the variables in direct variation, and \(k\) is the constant of variation, also known as the constant of proportionality. This constant represents the rate at which one variable changes in relation to the other.

To make the concept clearer for students, think of direct variation as a recipe. For every single unit of one ingredient, you need a fixed amount of another ingredient to maintain the taste of the dish. Here, the growth of the fingernail is directly proportional to the number of weeks passed, following a specific 'recipe' for growth per week.

Writing Equations

Writing equations is the process of translating words or real-world scenarios into mathematical language. It involves identifying the quantities involved and the relations between them. From our original exercise, we translate the statement 'fingernail growth varies directly as the number of weeks it has been growing' into the equation \(G = kW\). The direct variation relationship is now succinctly expressed as an equation.

When writing these equations, clarity is key. It's important to use consistent symbols for variables and to express constants explicitly. Once the equation is established, it can then be manipulated, such as substituting known values to solve for unknowns. In the context of teaching, using real-world examples like fingernail growth can help students understand the abstract concept by associating it with something tangible. This is an integral part of writing effective algebraic equations. Simplifying equations in steps and substituting values gradually can make the process even more digestible for students.

Constants of Variation

The constant of variation, denoted here as \(k\), plays a pivotal role in direct variations. It is the unchanging value that relates the variables in a direct variation equation to one another. In the exercise, \(k\) is identified as the rate of fingernail growth per week, which is 0.02 inches/week. This constant ensures that the relationship between \(G\) and \(W\) is linear and directly proportional.

Understanding the constant of variation allows us to make predictions and find unknown values. Once \(k\) is known, we can predict the growth for any number of weeks (\(W\)) by plugging it into the equation \(G = kW\). For educators, it's important to emphasize that the constant of variation does not change; it is the 'constant' part in any scenario described by a direct variation. By comprehensively explaining the constant of variation—perhaps through a variety of different examples—students can understand how to identify and apply it across different problems.