Problem 107 - College Algebra and Calculus: An Applied Approach

Question

The approximate lengths and diameters (in inches) of common nails are shown in the table. Find a logarithmic equation that relates the diameter \(y\) of a common nail to its length \(x\).$$ \begin{array}{|c|c|} \hline \begin{array}{c} \text { Length, } \\ x \end{array} & \begin{array}{c} \text { Diameter, } \\ y \end{array} \\ \hline 1 & 0.070 \\ \hline 2 & 0.111 \\ \hline 3 & 0.146 \\ \hline \end{array} $$$$ \begin{array}{|c|c|} \hline \begin{array}{c} \text { Length, } \\ x \end{array} & \begin{array}{c} \text { Diameter, } \\ y \end{array} \\ \hline 4 & 0.176 \\ \hline 5 & 0.204 \\ \hline 6 & 0.231 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The logarithmic equation that relates the diameter of a common nail to its length is \(y = -0.021 + 0.577 \cdot \ln(x)\). Please substitute any required lengths into this equation to find the corresponding diameters.

Step 1: Identify the data points

List each length-diameter pair as a coordinate point: (1, 0.070), (2, 0.111), (3, 0.146), (4, 0.176), (5, 0.204), (6, 0.231).

Step 2: Initial equation form

In general, a logarithmic function can be expressed as \(y=a+b \cdot \ln(x)\) where a and b are constants to be determined. We will use the given data to calculate the values of a and b.

Step 3: Calculate ‘b’

(b) can be calculated by finding the slope between two points in the data set. Here, we'll use the first two points (1, 0.070) and (2, 0.111). The slope can be found using the formula \(b=(y_2-y_1)/ (\ln(x_2) - \ln(x_1))\). So, \(b=(0.111-0.070)/ (\ln(2) - \ln(1))\) which simplifies to approximately 0.577.

Step 4: Calculate ‘a’

We find the constant (a) using the equation \(a=y-b \cdot \ln(x)\). Using point (1, 0.070) and the value for ‘b’ we calculated in Step 3, we get \(a= 0.070 - 0.577 \cdot \ln(1)\). This simplifies to approximately -0.021.

Step 5: Formulate final equation

Now we use the computed a and b values to form our final equation which results as \(y = -0.021 + 0.577 \cdot \ln(x)\).

Key Concepts

Logarithmic Equations

Logarithmic equations are a vital component in the field of algebra, especially when it comes to modeling growth patterns or measuring exponential decay. A logarithmic equation is, at its core, an equation that involves a logarithm. For instance, if we take the natural logarithm (denoted as \( \ln \)), a logarithmic relationship in algebra can be expressed in the form \( y = a + b \cdot \ln(x) \), where \( a \) and \( b \) are constants that must be determined based on provided data points. This form is particularly useful when we're trying to find a relationship between two variables, like the diameter and the length of nails in our example.
Understanding how to construct and solve logarithmic equations helps in making predictions and understanding the inherent relationships within different sets of data. With the exercise provided, the objective is to find a logarithmic equation that relates the diameter of a common nail to its length using the given data points. This involves identifying the data points, choosing a form for the equation, and calculating the constants using formulas derived from the logarithmic properties.

Data Points in Algebra

In algebra, data points are often represented as pairs of coordinates, such as \( (x, y) \) in the context of a two-dimensional graph. They are the basic units of information that provide quantitative values about relationships between variables. For the task at hand, each nail's length and diameter pair is a crucial data point that we analyze to establish a functional relationship through a logarithmic equation.
In our example, the data points are listed as \( (1, 0.070), (2, 0.111), (3, 0.146), (4, 0.176), (5, 0.204), (6, 0.231) \), which correlate the length of a nail to its diameter. Understanding how to use these data points effectively is an essential skill in algebra because it allows us to represent and analyze real-world situations mathematically. By plotting these data points on a graph, we can visualize the trend and pattern, which often helps in the determination of the exact formula that ties the data together.

Slope Calculation

Calculating the slope is fundamental in understanding the rate of change between two points on a graph. In algebra, the slope is often denoted as \( b \) and represents the steepness of a line, revealing how much one variable changes on average when another variable changes. The formula to calculate the slope between two points, \( (x_1, y_1) \) and \( (x_2,y_2) \), in the context of a logarithmic relationship, is \( b=(y_2-y_1)/ (\ln(x_2) - \ln(x_1)) \).
In the provided exercise, we calculate the slope as part of determining the constant \( b \) for our logarithmic equation. This is achieved by selecting two data points from our set and applying the slope formula. For the pair of points (1, 0.070) and (2, 0.111), we found that \( b \approx 0.577 \). Understanding slope calculation not only assists in solving this specific problem but also builds a foundation for studying more complex algebraic topics, such as functions and their graphs.