Problem 75 - Algebra A Combined Function

Question

Forensic scientists use the following functions to find the height of a woman if they are given the length of her femur bone \((f)\) or her tibia bone \((t)\) in centimeters. \(H(f)=2.59 f+47.24\) \(H(t)=2.72 t+61.28\) Use these functions to answer Exercises 75 and 76 Find the height of a woman whose femur measures 46 centimeters.

Step-by-Step Solution

Answer

The woman's height is 166.38 centimeters.

Step 1: Identify the Given Information

We are given the length of a woman's femur bone as 46 centimeters.

Step 2: Select the Appropriate Function

For the length of the femur bone, use the function \(H(f) = 2.59f + 47.24\).

Step 3: Substitute the Given Value into the Function

Substitute \(f = 46\) into the function: \(H(46) = 2.59 \times 46 + 47.24\).

Step 4: Perform the Calculation

Calculate \(2.59 \times 46 = 119.14\). Then add 47.24 to get \(H(46) = 119.14 + 47.24 = 166.38\).

Step 5: State the Result

The height of the woman, based on the femur measurement, is 166.38 centimeters.

Key Concepts

Forensic Science: Estimating Height

Forensic science is a fascinating field that helps solve mysteries using scientific principles. One of its useful applications is in estimating the height of an individual from their bone measurements. This can be pivotal in identifying unknown individuals in legal cases or archaeological findings.
Scientists have established correlations between specific bones and an individual's height.

• Femur and tibia bones are prominently used because of their relation to overall body height.
• Experts calculate height by measuring these bones and using mathematical formulas.

This provides a practical solution for situations where only skeletal remains are available. Understanding these connections forms the backbone of forensic anthropology and enhances the ability of forensic scientists to contribute crucial information in investigations.

Understanding Linear Functions

Linear functions form the foundation for many calculations, including those in forensic science. A linear function is a mathematical expression that shows a constant rate of change and is graphically represented as a straight line.
The general form of a linear function is given by:

• y = mx + b

where:

• y is the dependent variable (which we want to find).
• m is the slope or rate of change.
• x is the independent variable (the input value).
• b is the y-intercept (the value of y when x is zero).

In the context of height estimation, linear functions allow forensic scientists to predict height by inputting bone measurements. They use specific constants derived from statistical studies, which estimate an average outcome for various populations. These calculations are vital in scenarios where direct measurement of height isn't possible.

The Role of Bone Measurements

Bone measurements are key factors in estimating height and play a critical role in forensic investigations. The femur and tibia are often measured due to their strong correlation with full body height. The accuracy of these measurements allows scientists to estimate height effectively:

• The femur is the longest bone in the body, providing a robust basis for height estimation.
• The tibia complements the femur, offering additional data for more accurate predictions.

Measurements are typically taken in centimeters and plugged into predefined linear functions.
This process not only aids in estimating the height of deceased individuals but also assists in personal identification. As these bones are durable, they can be used even when other identification methods are not viable. Bone measurement methodologies, thus, enhance our understanding of past populations and contribute significantly to modern forensic practices.

Algebraic Calculation in Height Estimation

Algebraic calculation is central in translating bone measurements into height predictions through linear equations. The process involves straightforward arithmetic operations:

• Substitute the bone measurement into the designated formula.
• Multiply the measurement by the specified coefficient.
• Add the constant to compute the estimated height.

For example, if you know the length of a femur, like 46 centimeters, you use the equation:\( H(f) = 2.59f + 47.24 \)Plug in the femur measurement:\( H(46) = 2.59 imes 46 + 47.24 \)First, compute \( 2.59 imes 46 = 119.14 \).Then, add \( 47.24 \) to get a total of \( 166.38 \) centimeters.
This straightforward calculation showcases how algebra makes sense of data, delivering valuable insights with clarity. Algebra serves as a bridge in transforming raw measurements into comprehensible information, essential in forensic analysis and beyond.