Problem 123 - College Algebra with Modeling and Visualization

Question

Ring Size \(\quad\) The table lists ring size \(S\) for a finger with circumference \(x\) in centimeters. $$ \begin{array}{ccccc} x(\mathrm{cm}) & 4.65 & 5.40 & 5.66 & 6.41 \\ S(\text { size }) & 4 & 7 & 8 & 11 \end{array} $$ Source: Overstock (a) Find a linear function \(S\) that models the data. (b) Find the circumference of a finger with a ring size of 6

Step-by-Step Solution

Answer

The linear function is \( S = 3.96x - 10.394 \). The circumference for size 6 is approximately 4.14 cm.

Step 1: Identify two points to calculate slope

First, select two points from the data table to calculate the slope of the linear function. Let's choose the points \((4.65, 4)\) and \((5.66, 8)\). These points correspond to \(x\) values of 4.65 cm and 5.66 cm and ring sizes \(S\) of 4 and 8 respectively.

Step 2: Calculate the slope of the line

The slope \(m\) of a line is found using the formula \((y_2 - y_1) / (x_2 - x_1)\). Substitute the \(x\) and \(S\) values from Step 1: \[ m = \frac{8 - 4}{5.66 - 4.65} = \frac{4}{1.01} \approx 3.96 \]

Step 3: Write the point-slope form of the equation

Now use one of the points and the slope to write the equation using the point-slope form \(y - y_1 = m(x - x_1)\). Using point \(4.65, 4\) and the calculated slope, the equation becomes \[ S - 4 = 3.96(x - 4.65) \]

Step 4: Solve for S to find the linear model

Rearrange the equation from Step 3 into slope-intercept form \(y = mx + b\). This gives \[ S = 3.96x - (3.96 \times 4.65) + 4 \] Simplify to find the linear function: \[ S = 3.96x - 14.394 + 4 = 3.96x - 10.394 \].

Step 5: Use the linear function to solve for x when S = 6

To find the circumference \(x\) for a ring size or \(S = 6\), substitute 6 into the equation: \[ 6 = 3.96x - 10.394 \] Solve for \(x\): \[ 6 + 10.394 = 3.96x \] \[ 16.394 = 3.96x \] \[ x = \frac{16.394}{3.96} \approx 4.14 ext{ cm} \]

Key Concepts

Slope Calculation

When we talk about the slope of a line, we are referring to its steepness or inclination. To find it, you can use two points from the line. The formula used is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points.

In this exercise, we use the points \((4.65, 4)\) and \((5.66, 8)\). Plugging these values into the formula gives:

• \( y_2 - y_1 = 8 - 4 = 4 \)
• \( x_2 - x_1 = 5.66 - 4.65 = 1.01 \)

Thus, the slope \( m \) is \( \frac{4}{1.01} \approx 3.96 \). This means that for every centimeter increase in circumference, the ring size increases by about 3.96 units.

Point-Slope Form

The point-slope form of a line's equation is a useful way to represent a linear function. It's written as \( y - y_1 = m(x - x_1) \). Here, \((x_1, y_1)\) are the coordinates of a point on the line, and \(m\) is the slope.

Using the point \((4.65, 4)\) and the calculated slope \(3.96\), we establish the equation:

• \( S - 4 = 3.96(x - 4.65) \)

This form makes it easy to see how changes in \(x\) affect \(S\), starting from a known point.

Linear Model

To make predictions, we transform the point-slope formula into a linear equation, i.e., \( y = mx + b \), where \(y\) is the dependent variable, \(x\) is the independent variable, \(m\) is the slope, and \(b\) is the y-intercept.

From the equation \( S - 4 = 3.96(x - 4.65) \), rearrange to solve for \(S\):

• Expand: \( S = 3.96x - 18.414 + 4 \)
• Simplify: \( S = 3.96x - 14.394 \)

This linear model, \( S = 3.96x - 14.394 \), allows us to estimate the ring size based on the circumference. It shows the relationship between ring size \(S\) and finger circumference \(x\).

Circumference Calculation

Circumference is the linear distance around a circular object. In this problem, we calculate the finger circumference related to a specific ring size using our linear model.

To find the circumference when \(S = 6\), substitute into the model equation \(S = 3.96x - 14.394\):

• Set \(S = 6\): \(6 = 3.96x - 14.394\)
• Resolve for \(x\):
  • Add 14.394 to both sides: \(20.394 = 3.96x\)
  • Divide by 3.96: \(x = \frac{20.394}{3.96} \approx 5.15\)

So, the circumference for a ring size 6 is approximately 5.15 cm. This method uses the linear model to back calculate unknown values from known outputs.