Problem 85 - Algebra A Combined Function

Question

Anthropologists and forensic scientists use a measure called the cephalic index to help classify skulls. The cephalic index of a skull with width \(W\) and length \(L\) from front to back is given by the formula $$ C=\frac{100 W}{L} $$ A long skull has an index value less than \(75,\) a medium skull has an index value between 75 and \(85,\) and a broad skull has an index value over \(85 .\) Find the cephalic index of a skull that is 5 inches wide and 6.4 inches long. Classify the skull.

Step-by-Step Solution

Answer

The cephalic index is approximately 78.125, classifying the skull as medium.

Step 1: Identify Known Values

We are given that the width \(W\) of the skull is 5 inches and the length \(L\) is 6.4 inches. These will be our inputs for the cephalic index formula.

Step 2: Write Down the Cephalic Index Formula

The cephalic index \(C\) is calculated using the formula \( C = \frac{100 W}{L} \). We will use this formula to find the cephalic index for the given skull.

Step 3: Substitute the Values into the Formula

Substitute the given values into the formula: \( C = \frac{100 \times 5}{6.4} \). This will allow us to calculate the cephalic index.

Step 4: Perform the Calculation

Calculate the cephalic index: \( C = \frac{100 \times 5}{6.4} = \frac{500}{6.4} \). Carry out the division to find \(C\).

Step 5: Simplify the Result

Simplify \( \frac{500}{6.4} \) to get \( C \approx 78.125 \). This is the cephalic index of the skull.

Step 6: Classify the Skull

Using the classification criteria: - Long skull: index < 75- Medium skull: 75 \( \leq \) index \( \leq \) 85- Broad skull: index > 85Since \( C = 78.125 \), the skull is classified as a medium skull.

Key Concepts

Forensic Anthropology

Forensic anthropology is a specialized field that combines anthropology and forensic science to help identify human remains. It plays a crucial role in criminal investigations by assisting in determining the age, sex, ancestry, and health status of deceased individuals from their bones. In particular, forensic anthropologists focus on skeletal features to reconstruct biological profiles and contribute to solving crimes.
This field heavily relies on understanding human skeletal variations. Such variations can help solve cases by determining if a skull belongs to a particular demographic group. This is where techniques like classification of skulls using the cephalic index become incredibly useful. Forensic anthropology combines scientific methods to interpret findings, making it an ideal partner in crime scene investigations.

Skull Measurement

Skull measurement is a fundamental practice in anthropology and forensic science for classifying and studying human populations. Measurements help experts assess skull shapes, sizes, and other features. Key measurements include the width and length of the skull, which are crucial for calculating the cephalic index.
The cephalic index specifically looks at the ratio of skull width to length, which reveals information about the skull shape:

• Width (W): Typically measured at the widest point of the parietal bones.
• Length (L): Measured from the most forward point of the frontal bone to the furthest back point of the occipital bone.

By understanding these measurements, anthropologists can compare skulls across different populations, providing insights into evolutionary and genetic relationships.

Mathematical Classification

Mathematical classification involves creating categories based on quantitative measurements, which simplifies the analysis of complex data. In the context of skulls, it turns anthropological data into easily understandable classifications.
The cephalic index is one such mathematical classification, dividing skulls into three main types based on their shape:

• Long skulls: Index less than 75
• Medium skulls: Index between 75 and 85
• Broad skulls: Index greater than 85

This classification allows anthropologists to quickly identify and compare skull types by using a single numerical value. It provides a straightforward way to categorize physical traits and understand broad patterns in anthropometric studies.

Index Calculation

Index calculation is a method used to express the relationship between two measurements as a single number. The cephalic index is a classic example, where it provides a ratio of two skull dimensions. This calculation is crucial for classifying skulls based on their width and length.
To calculate the cephalic index, follow these steps:

• Identify the width (W) and length (L) of the skull.
• Use the formula for the cephalic index: \( C = \frac{100 \times W}{L} \).
• Substitute the known values into the formula.
• Perform the division and obtain the cephalic index value.

For example, with a skull width of 5 inches and length of 6.4 inches, substitution into the formula yields \( C \approx 78.125 \), identifying it as a medium skull. This systematic approach ensures consistent and clear results when classifying skull shapes.