Problem 50 - 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.) From the foot bones of an adult human male, an anthropologist estimates that the person's height was 175 centimeters. A few feet away from the site where the foot bones were discovered, the anthropologist discovers a male adult femur that is 48 centimeters long. Is it likely that both the foot bones and the thigh bone came from the same person?

Step-by-Step Solution

Answer

The answer depends on the calculation in step 2 and the comparison with the given height of 175 cm in step 3. A conclusion on whether the bones likely belong to the same person can only be drawn from this comparison.

Step 1: Identification of Relevant Formulas

Given are the equations for estimating the height based on the femur length, separately for males and females. Since it is already known that the bones belong to an adult male, use the male equation: \(y=0.442x - 29.37\) where \(y\) is the femur length and \(x\) is the height.

Step 2: Application of Formula

Re-arrange the formula to calculate for height \(x\), given that the femur length \(y\) is 48 cm. This results in the formula \(x = (y+29.37)\) / 0.442. Substitute \(y = 48\) cm into the formula to estimate the height of the person the femur bone belongs to.

Step 3: Comparison and Conclusion

Compare the height calculated from the femur length to the given height from the foot bones of 175 cm. If they are quite close, it can be concluded that it is likely the two sets of bones are from the same person. A difference beyond a certain threshold might suggest they are likely from different individuals. The threshold is subjective but can be determined based on the accuracy required.

Key Concepts

Femur Length to Height Estimation

Understanding the relationship between the femur length and the height of an individual is essential in anthropology, particularly when assessing skeletal remains. The femur is the longest bone in the human body and serves as a critical indicator in estimating a person's stature. This estimation is based on regression equations derived from population statistics that correlate femur length with total body height.

For practical purposes, two separate linear equations have been developed to approximate the height of adult males and females, taking into account gender differences. These formulas, expressed as y = 0.386x - 19.20 for females and y = 0.442x - 29.37 for males, enable anthropologists to back-calculate the approximate height (x) from a given femur length (y). It's important to use the correct equation according to the sex of the individual to avoid any inaccuracies.

By analyzing the dimensions of the femur, conclusions can be drawn about the height of the deceased, which can then be compared to other skeletal remains or demographic data. This analysis serves a dual purpose: to help identify unknown individuals and to provide insights into the living conditions and health of past populations.

Solving Linear Equations

Linear equations are foundational to numerous fields, including anthropology. They are used to model relationships between two variables, typically in the form of y = mx + b, where m is the slope, representing the change in y for each unit increase in x, and b is the y-intercept, the value of y when x equals zero.

When solving a linear equation to find an unknown variable, manipulation of the equation by isolating the unknown on one side is a common approach. This might involve simple operations such as addition, subtraction, multiplication, or division. For instance, in the anthropological context, if the femur length (y) is known, the equation can be rearranged to solve for the height (x), as shown in Step 2 of the provided solution: x = (y+29.37) / 0.442.

The ability to solve linear equations effectively is important not only for anthropological studies but also for many real-world applications. Accurate and careful manipulation of these equations is critical to ensure reliable results and meaningful interpretations.

Anthropological Forensics

Anthropological forensics plays a significant role in modern-day forensic science, often bridging the gap between archaeological methods and the legal investigation of death. Forensic anthropologists apply knowledge of human osteology, or the study of bones, to address questions of identification and analysis of human remains.

The process encompasses a thorough examination of skeletal remains, determining characteristics such as age, sex, stature, and any signs of trauma or disease. This information can be vital in missing persons cases, mass disaster scenarios, and criminal investigations.

In our exercise, the anthropologist uses a linear equation to estimate the height of the individual from the femur length. This is a typical task for a forensic anthropologist, helping to determine whether different bones might have originated from the same individual. Such estimations, when supplemented with other anthropological evidence, can significantly contribute to case resolutions. The precise and respectful analysis of bones not only aids in identification but also tells stories that would otherwise remain untold, providing closure to families and adding details to the tapestry of human history.