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 \(y=0.432 x-10.44\) \(y=0.449 x-12.15\) where \(y\) is the length of the femur in inches and \(x\) is the height of the adult in inches (see figure). From the foot bones of an adult human male, an anthropologist estimates that the male was 65 inches tall. A few feet away from the site where the foot bones were discovered, the anthropologist discovers an adult male femur that is 17 inches long. Is it possible that the leg and foot bones came from the same person? Explain.

Short answer

No, it is not 100% certain that the leg and foot bones came from the same person as neither of the two equations give an exact femur length of 17 inches for a height of 65 inches, even though both calculated lengths are relatively close.

Step-by-step solution

Substituting the given height into the first equation

By substituting \(x = 65\) into the first equation \(y=0.432x -10.44\) we will find the length of femur according to this equation. This should result in \(y = 0.432*65 -10.44 = 17.88\) inches.

Substituting the given height into the second equation

By substituting \(x = 65\) into the second equation \(y=0.449x -12.15\) we will find the length of femur according to this equation. This should result in \(y = 0.449*65 -12.15 = 17.035\) inches.

Compare the calculated femur lengths to the actual

Now it's essential to compare the two calculated femur lengths to the given 17 inches. The length from the first equation is 17.88 inches, and for the second one, it is 17.035 inches. Neither of them exactly match the given femur length.

Make a Conclusion

On comparison, while both estimates are relatively close to the actual length of 17 inches, neither are exact. Although it is not impossible, it would be unlikely that the femur and foot bones came from the same person.

Key concepts

College Algebra

In the realm of college algebra, one often deals with equations that model real-world situations. Algebra is known for its symbolic manipulation to solve for unknown values, and it serves as a critical foundation for many branches of science and business. Understanding how to formulate and rearrange equations is a vital skill that encapsulates much of college-level algebra.

A core component of college algebra is dealing with variables and constants to express relationships. In this particular exercise, the relationship between the length of a femur and an individual's height is captured algebraically. The equations provided are simple linear equations, which are introductory yet essential algebraic constructs. They express a direct proportionality between two quantities where one quantity is a constant multiple of the other, combined with a constant term, demonstrating the core algebraic principle of functions and their applications in real-life scenarios.

Solving Linear Equations

The process of solving linear equations involves finding the value of the variable that makes the equation true. This is commonly done by isolating the variable on one side of the equation. Linear equations are typically in the form of \(y = mx + b\), where \(m\) represents the slope, \(b\) is the y-intercept, and \(x\) and \(y\) are variables.

In the given exercise, the linear equations provided allow us to estimate the femur length based on a person's height. Applying substitution is one of the most straightforward methods to solve these types of problems. For instance, substituting \(x = 65\) into the first equation helps us calculate the estimated length of femur for a person who is 65 inches tall. This exemplifies how algebraic skills are employed to draw insights from equations that model real-world phenomena.

Mathematical Modeling

The concept of mathematical modeling is integral in translating real-world situations into a mathematical framework that allows for analysis and predictions. A model uses mathematical language to describe a system or process, often simplifying complex phenomena to their essential characteristics represented by variables and equations.

In this scenario, mathematical models have been used to approximate the relationship between human height and femur length. The anthropologist has two linear equations at disposal, reflecting diverse data or possibly different population samples. By utilizing these models with the known height, the femur length is estimated. This is powerful because it demonstrates how models are used to predict outcomes based on given inputs. Additionally, mathematical models enable us to assess the likelihood of events, like determining if the discovered bones belong to the same individual, enhancing our comprehension of the world through the lens of mathematics.