Problem 40 - Algebra and Trigonometry with Analytic Geometry

Question

Archeologists can determine the height of a human without having a complete skeleton. If an archeologist finds only a humerus, then the height of the individual can be determined by using a simple linear relationship. (The humerus is the bone between the shoulder and the elbow.) For a female, if \(x\) is the length of the humerus (in centimeters), then her height \(h\) (in centimeters) can be determined using the formula \(h=65+3.14 x\). For a male, \(h=73.6+3.0 x\) should be used. (a) A female skeleton having a 30 -centimeter humerus is found. Find the woman's height at death. (b) A person's height will typically decrease by \(0.06\) centimeter each year after age 30 . A complete male skeleton is found. The humerus is 34 centimeters, and the man's height was 174 centimeters. Determine his approximate age at death.

Step-by-Step Solution

Answer

The woman's height was 159.2 cm and the man's age at death was 57 years.

Step 1: Calculating Height of Female

Start by determining the woman's height using the given linear formula for a female: \( h = 65 + 3.14x \). Substitute \( x = 30 \) into the equation: \[h = 65 + 3.14 \times 30\] This calculates to:\[h = 65 + 94.2 = 159.2\text{ cm}.\]

Step 2: Setting Up Age Decrease Equation

For the male skeleton, we know the formula to find height from the humerus length and the decrease in height over age. Use the male height formula: \( h = 73.6 + 3.0x \). Calculate the expected height from the humerus when first forensic growth completed by using: \[h = 73.6 + 3.0 \times 34 = 175.6\text{ cm}.\]

Step 3: Calculating Age at Death

Determine the difference between the initial calculated height and the found height:\[\text{Initial height} - \text{Height at death} = 175.6 - 174 = 1.6\text{ cm}\]Divide this difference by the annual height decrease \(0.06\text{ cm/year}\) to estimate the years after 30:\[\text{Age difference} = \frac{1.6}{0.06} \approx 26.67 \text{ years}\]This suggests the man was approximately \(30 + 27 = 57\) years old when he died.

Key Concepts

Forensic Anthropology

Forensic Anthropology is a fascinating field that combines the principles of anthropology with forensic science. It's used to identify human remains and understand the history and circumstances of those individuals' lives. This field plays a crucial role in archeology, crime scene investigations, and even natural disaster responses. By analyzing bones, forensic anthropologists can determine age, sex, ancestry, and stature of deceased individuals. The physical characteristics of bones offer clues that help reconstruct how people lived and what may have happened to them. In cases where a full skeleton is not available, such as finding only a humerus, forensic anthropologists use mathematical formulas to estimate characteristics like height. This estimation is based on well-studied averages and linear relationships between bone lengths and total height. Forensic anthropology thus blends scientific inquiry with mathematical precision to solve mysteries of the past.

Linear Relationships

Linear Relationships are mathematical connections where individual elements change consistently with each other. In the context of bones and height, a linear equation helps predict one measurement based on another. Such a relationship is often graphed as a straight line, showing the constant rate of change between the variables. For instance, in the problem provided, linear equations express the relationships between arm bone length and an individual’s height. For females, the equation is given by \( h = 65 + 3.14x \) and for males \( h = 73.6 + 3.0x \). Here, \( h \) represents height and \( x \) the humerus length. These equations allow forensic anthropologists to make educated guesses about height using a straightforward mathematical model. Understanding and applying linear relationships simplify complex human biological data into manageable equations.

Bone Length to Height Estimation

Bone Length to Height Estimation is an essential technique in forensic anthropology. It allows for the estimation of a person's height when only partial skeletal remains, like a single bone, are available. The equations \( h = 65 + 3.14x \) for females and \( h = 73.6 + 3.0x \) for males are practical tools used to determine height from the humerus length.

Female Height Calculation:

Suppose a female skeleton with a 30 cm humerus is found. Using the equation \( h = 65 + 3.14x \):

• Substitute \( x = 30 \) into the equation.
• Calculate \( 65 + 3.14 \times 30 = 159.2 \). Hence, her height was approximately 159.2 cm.

Estimating Age Using Height Changes:

For a male, you might find a situation where the height at death is compared against the calculated height from the humerus length.

• Use the formula: \( h = 73.6 + 3.0 \times 34 = 175.6 \ text{ cm} \).
• Subtract the height at death from this to find the height lost with age: \( 175.6 - 174 = 1.6 \text{ cm} \).
• Considering a decrease of 0.06 cm/year after age 30, divide 1.6 cm by 0.06 cm/year to find the years over 30.
• This calculation can estimate the age at death, aiding forensic analysis.

By applying these linear equations, one can convert simple bone measurements into significant scientific insights.