Problem 31 - College Algebra

Question

A person's fingernail growth, \(G\), in inches, varies directly as the number of weeks it has been growing, \(W\). a. Write an equation that expresses this relationship. b. 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 length 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

The equation expressing the relationship is \(G = kW\). Substituting 0.02 for \(k\), the equation becomes \(G = 0.02W\). Finally, substituting 52 for \(W\) the length of the fingernail at the end of one year would be \(G = 0.02 * 52\) or \(G = 1.04\) inches.

Step 1: Write a Direct Variation Equation

The direct variation can be modeled mathematically with the equation \(G = kW\), where \(G\) is the fingernail growth in inches, \(W\) is the number of weeks, and \(k\) is the constant of variation.

Step 2: Substituting given value for constant of variation

From the problem, it's given that the constant of variation, \(k\), is 0.02 inch per week. The new equation would then be \(G = 0.02W\), where \(G\) is the growth and \(W\) is the number of weeks.

Step 3: Predicting fingernail growth after a year

Substituting 52 (representing 52 weeks in a year) for \(W\), the equation becomes \(G = 0.02 * 52\). Solving this multiplication gives the fingernail growth after a year.

Key Concepts

Fingernail Growth

Fingernails, much like other parts of our body, are continuously growing. Growth varies depending on several factors like age, diet, and overall health, but in this scenario, we assume a constant growth rate. Direct variation helps us easily determine fingernail growth over a period of time.
This means that the length fingernails can grow is directly dependent on how long you let them grow without cutting or breaking them. For example, if you choose not to cut your nails for one year, their growth can be calculated directly through the number of weeks in that year.

• The growth is measured in inches.
• In our exercise, nails grow at a rate of 0.02 inches per week.
• We can easily extend this rate to find the total growth over any number of weeks.

Using a mathematical equation allows these variables to be manipulated efficiently.

Constant of Variation

In our exercise, the constant of variation is crucial. It is represented by the symbol \(k\). This constant defines the rate at which fingernails grow per week. Here, it is given as 0.02 inches per week. This means for every week, your fingernails grow 0.02 inches.
To apply this in a real-world context, consider how consistent nail growth can affect nail length:

• If \(k\) increases, nails grow faster each week.
• If \(k\) is smaller, nails grow slower.

Understanding the constant of variation helps not only in predicting weekly growth but also in identifying abnormal growth patterns if \(k\) changes.

Mathematical Modeling

Mathematical modeling involves using equations to represent real-world phenomena. In the context of fingernail growth, we express the growth using the equation \(G = kW\).
This equation exemplifies a direct variation, where one quantity is a constant multiple of another. Here are the components:

• \(G\) stands for total growth, measured in inches.
• \(k\) is the constant rate of growth per week.
• \(W\) is the number of weeks over which growth is measured.

This model makes predictions straightforward: by substituting in values for \(k\) and \(W\), anyone can calculate expected nail growth. For instance, substituting \(k = 0.02\) and \(W = 52\) for a whole year gives us \(G = 0.02 \times 52 = 1.04\). Therefore, without interruptions, nails can potentially grow about 1.04 inches in a year.