Forensic scientists use the lengths of certain bones to...

Forensic scientists use the lengths of certain bones to calculate the height of a person. Two such bones are the tibia $(t),$ the bone from the ankle to the knee, and the femur $(r),$ the bone from the knee to the hip socket. A person's height $(h)$ in centimeters is determined from the lengths of these bones using the following functions. For men: $\quad h(r)=69.09+2.24 r$ or $\quad h(t)=81.69+2.39 t$ For women: $\quad h(r)=61.41+2.32 r$ or $\quad h(t)=72.57+2.53 t$ (a) Find the height of a man with a femur measuring $56 \mathrm{~cm}$. (b) Find the height of a man with a tibia measuring $40 \mathrm{~cm}$. (c) Find the height of a woman with a femur measuring $50 \mathrm{~cm}$. (d) Find the height of a woman with a tibia measuring $36 \mathrm{~cm}$.

Forensic scientists use the lengths of certain bones to calculate the height of a person. Two such bones are the tibia $(t),$ the bone from the ankle to the knee, and the femur $(r),$ the bone from the knee to the hip socket. A person's height $(h)$ in centimeters is determined from the lengths of these bones using the following functions.
For men: $\quad h(r)=69.09+2.24 r$ or $\quad h(t)=81.69+2.39 t$
For women: $\quad h(r)=61.41+2.32 r$ or $\quad h(t)=72.57+2.53 t$
(a) Find the height of a man with a femur measuring $56 \mathrm{~cm}$.
(b) Find the height of a man with a tibia measuring $40 \mathrm{~cm}$.
(c) Find the height of a woman with a femur measuring $50 \mathrm{~cm}$.
(d) Find the height of a woman with a tibia measuring $36 \mathrm{~cm}$.

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Key Concepts

Linear Functions

Linear functions describe relationships with a constant rate of change and are expressed in the form y = mx + b, where m is the slope and b is the y-intercept. They are widely used in applied mathematics to model phenomena where one quantity linearly depends on another, such as estimating a person's height based on bone measurements.

Slope Interpretation

The slope in a linear function represents the rate at which the dependent variable changes with respect to the independent variable. In the context of height estimation, the slope indicates how much a person’s height is expected to increase per unit increase in the length of a specific bone.

Y-Intercept Concept

The y-intercept is the value of the dependent variable when the independent variable is zero. In these models, the y-intercept can be interpreted as a baseline height, accounting for additional factors beyond the measured bone length that contribute to a person's overall height.

Function Evaluation

Function evaluation involves substituting a specific value of the independent variable into the function to calculate the corresponding value of the dependent variable. This technique is essential for applying mathematical models to real-world data, such as determining a person's height when given a particular bone measurement.

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