Problem 49 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the following information. The relationship between the length of an adult's femur (thigh bone) and the height of the adult can be approximated by the linear equations $$y=0.386 x-19.20 \quad \text { Female }$$ $$y=0.442 x-29.37 \quad \text { Male }$$ where \(y\) is the length of the femur in centimeters and \(x\) is the height of the adult in centimeters. (See figure.) An anthropologist discovers a femur belonging to an adult human female. The bone is 43 centimeterss long. Estimate the height of the female.

Step-by-Step Solution

Answer

The estimated height of the female is around 161 cm.

Step 1 - Identify the correct equation

The femur is from a female, so we need to use the equation for females: \(y=0.386x-19.20\). Here, \(y\) represents the length of the femur and \(x\) represents the height of the female.

Step 2 - Substitute the value

We know that the length of the femur (\(y\)) is 43 cm. Substitute \(y=43\) into the equation and solve for \(x\), which becomes \(43=0.386x-19.20\).

Step 3 - Solve for x

First, isolate 'x' by adding 19.20 to both sides of the equation, which gives \(43+19.20=0.386x\). Then divide both sides by 0.386 to solve for \(x\). Calculating this gives an \(x\) value of approximately 161 cm.

Key Concepts

Solving Linear Equations

Solving a linear equation is akin to finding the missing piece of a puzzle. In our everyday life, a linear equation can represent anything that has a constant rate of change, similar to the way speed relates to time in a moving vehicle. In the context of our exercise, the equation represents the relationship between the height and femur length of an individual.

The process involves isolating the variable we want to find - in this case, the height of the adult female (\(x\)) - by performing operations that maintain the equilibrium of the equation. We substitute known values into the equation and use algebraic operations to find the unknown variable. It's crucial to follow a methodical approach: identify the correct equation, substitute the values, and perform algebraic operations to solve for the variable of interest, just as demonstrated in the provided steps.

Applications of Linear Algebra

Linear algebra is not confined to the classroom; it has far-reaching applications in fields as diverse as computer graphics, engineering, and even in our study of the natural world. When it comes to anthropology, linear algebra is instrumental for making inferences about past lives.

By establishing relationships between different measurements through equations, anthropologists can predict missing data, project population trends, or even trace the journey of human evolution. For our exercise, we apply a linear equation to estimate a person's height from the length of the femur, which would otherwise have been a challenge using direct measurements, especially in cases involving incomplete skeletal remains.

Algebra in Anthropology

In anthropology, algebra becomes a powerful tool that helps to unlock the stories bones tell us. By using linear equations, anthropologists can estimate physical characteristics of individuals from the past, including their height, which is a vital piece of demographic information.

This information, in turn, can suggest aspects of the individual's health, lifestyle, and the environmental conditions they lived in. The linear equations used are derived from statistically significant data and can vary based on the population being studied as well as the differences that exist between sexes, as highlighted by the distinct formulas for females and males in our exercise.

Estimating Height from Bone Length

The practice of estimating height from bone length, particularly using the femur, is based on the principle that certain key dimensions of the human body maintain consistent ratios. Determining a person's height from the length of their femur involves understanding these ratios and applying them through linear equations.

Such estimations can play a critical role in forensic investigations and archaeological studies to help create biological profiles of individuals. For students learning about these practical applications, grasping the concept of linear equations is not only about solving mathematical problems but also about appreciating their real-world implications, as in the case of estimating the height of an individual through the length of a femur bone.