Problem 106 - Introductory and Intermediate Algebra for College Students 4th

Question

A woman'sheight, \(h\), is related to the length of her femur, \(f\) (the bone from the knee to the hip socket), by the formula \(f=0.432 h-10.44 .\) Both \(h\) and \(f\) are measured in inches. A partial skeleton is found of a woman in which the femur is 16 inches long. Police find the skeleton in an area where a woman slightly over 5 feet tall has been missing for over a year. Can the partial skeleton be that of the missing woman? Explain.

Step-by-Step Solution

Answer

Yes, the skeleton could be that of the missing woman since the calculated height from the femur length is 61 inches, which is slightly over 5 feet. Further investigations should be conducted for confirmation.

Step 1: Identify Given Data

The length of the femur found is given as 16 inches. It is also given that the missing woman is slightly above 5 feet tall, so her height in inches would be a bit above 60 inches.

Step 2: Substitute the given femur length into the formula

Substitute the value of femur \(f=16\) in the given formula, \(f=0.432 h-10.44\). This gives us \(16=0.432h -10.44\).

Step 3: Solve for height, \(h\)

In the equation, isolate \(h\) by adding \(10.44\) to both sides, and then dividing by \(0.432\). This will yield \(h\) as approximately \(61\) inches.

Key Concepts

Linear Equations

Linear equations are essential in understanding relationships between variables. These equations describe a straight line when plotted on a graph. The general form of a linear equation is \( y = mx + b \), where \( m \) represents the slope and \( b \) is the y-intercept. In our given problem, we have an equation relating height \( h \) to femur length \( f \): \( f = 0.432h - 10.44 \). Here, \( 0.432 \) is analogous to the slope, showing how changes in height affect the femur length, and \(-10.44\) is like the y-intercept. This equation shows a direct linear relationship, so by knowing one variable, we can find the other using simple algebraic manipulations. Linear equations are useful in many real-world applications, like calculating distances, predicting costs, or, as in this exercise, determining the relationship between different physical attributes of a person.

Problem Solving

Problem-solving in mathematics involves a step-by-step approach to finding solutions, much like solving a puzzle. It starts with understanding the problem and gathering the given information. In our exercise, we know the femur length of the skeleton \( f = 16 \text{ inches} \) and the estimated height of the missing woman as slightly over 5 feet (or more than 60 inches). The next step is to apply the known data to the equation \( f = 0.432h - 10.44 \) to check if the skeleton could belong to the missing woman. By substituting \( f = 16 \text{ inches} \) into the equation and solving for \( h \), we determine the height. Problem-solving requires logical reasoning, where each step logically follows from the previous one, leading to the correct conclusion. In this scenario, reaching \( h \approx 61 \text{ inches} \) helps verify that the skeleton might belong to the woman who was slightly taller than 5 feet.

Mathematical Formulas

Mathematical formulas are powerful tools that help us solve problems efficiently. The formula in our exercise \( f = 0.432h - 10.44 \) is specifically used to express the direct linear relationship between the femur length and a woman's height. To use this formula effectively, it is crucial to understand the roles of each term: \( 0.432h \) represents the proportional part indicating how much of the height contributes directly to femur length, and \(-10.44\) accounts for adjustments needed based on average anatomical proportions. When we substitute the value of \( f = 16 \) and solve for \( h \), we use basic algebraic techniques such as addition and division to isolate and calculate the unknown variable \( h \). Understanding these mathematical formulas enhances problem-solving capabilities, making it possible to derive meaningful conclusions from seemingly unrelated data.