Chapter 1.FOM, Problem 1P

Femur Length and Height Anthropologists use a linear model that relates femur length to height. The model allows an anthropologist to determine the height of an individual when only a partial skeleton (including the femur) is found. In this problem we find the model by analyzing the data on femur length and height for the eight males given in the table.

(a) Make a scatter plot of the data.

(b) Find and graph a linear function that models the data.

(c) An anthropologist finds a femur of length $58$ cm. How tall was the person?

Femur length (cm) Height (cm) 50.1 178.5 48.3 173.6 45.2 164.8 44.7 163.7 44.5 168.3 42.7 165.0 39.5 155.4 38.0 155.8

To determine

(a)

To make:

A scatter plot of the data given below:

Femur length (cm)	Height (cm)
50.1	178.5
48.3	173.6
45.2	164.8
44.7	163.7
44.5	168.3
42.7	165.0
39.5	155.4
38.0	155.8

Answer to Problem 1P

Solution:

Explanation of Solution

Approach:

A mathematical model is a mathematical representation of an equation or process. Mathematical models are used to predict or estimate the required output.

Scatter plots are the two dimensional graph that represent the coordinate values of the plotted points.

Calculation:

Given:

Femur length (cm)	Height (cm)
50.1	178.5
48.3	173.6
45.2	164.8
44.7	163.7
44.5	168.3
42.7	165.0
39.5	155.4
38.0	155.8

Taking the femur length as $x$-axis and height as $y$-axis, we plot the given points to obtain a scatter plot of the data as shown below:

To determine

(b)

To find:

The linear function that models the data and graph the function of the scatter plot.

Answer to Problem 1P

Solution:

The linear function of the data is found as $y=1.88x+82.65$.

Explanation of Solution

Approach:

A mathematical model is a mathematical representation of an equation or process. Mathematical models are used to predict or estimate the required output.

Scatter plots are the two dimensional graph that represent the coordinate values of the plotted points.

Calculation:

Given:

Femur length (cm)	Height (cm)
50.1	178.5
48.3	173.6
45.2	164.8
44.7	163.7
44.5	168.3
42.7	165.0
39.5	155.4
38.0	155.8

The scatter plot and the graph of the linear function that models the data are plotted as shown below:

The linear function of the data is found as $y=1.8807x+82.65$.

To determine

(c)

To find:

The height of the femur of length $58$ cm from the model of the function.

Answer to Problem 1P

Solution:

The height of the femur was $191.77$ cm for a length of $58$ cm.

Explanation of Solution

Approach:

A mathematical model is a mathematical representation of an equation or process. Mathematical models are used to predict or estimate the required output.

Scatter plots are the two dimensional graph that represent the coordinate values of the plotted points.

Calculation:

From the data, we find that function is $y=1.8807x+82.65$.

Substituting the value of $x=58$, in the above equation, we get

$y=1.8807x+82.65$

$=1.8807\left(58\right)+82.65$

$=191.73$

Thus, the height of the femur was $191.73$ cm for a length of $58$ cm.