91Ó°ÊÓ

Solve each problem. Forensic scientists use the lengths of certain bones to calculate the height of a person. Two bones often used 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 by using functions defined by the following formulas. For men: \(h(r)=69.09+2.24 r\) or \(h(t)=81.69+2.39 t\) For women: \(h(r)=61.41+2.32 r\) or \(h(t)=72.57+2.53 t\) 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 \(36 \mathrm{cm} .\) (PICTURE CANT COPY)

Step by step solution

01

Identify the formula for finding a man's height using the femur

Use the formula for men's height based on the femur length: \[ h(r) = 69.09 + 2.24r \]

02

Substitute the femur length into the formula

Substitute the given femur length of 56 cm into the formula. \[ h(56) = 69.09 + 2.24 \times 56 \]

03

Calculate the height for the given femur length

Perform the calculation: \[ h(56) = 69.09 + 125.44 = 194.53 \] The height of the man is 194.53 cm.

04

Identify the formula for finding a man's height using the tibia

Use the formula for men's height based on the tibia length: \[ h(t) = 81.69 + 2.39t \]

05

Substitute the tibia length into the formula

Substitute the given tibia length of 40 cm into the formula. \[ h(40) = 81.69 + 2.39 \times 40 \]

06

Calculate the height for the given tibia length

Perform the calculation: \[ h(40) = 81.69 + 95.6 = 177.29 \] The height of the man is 177.29 cm.

07

Identify the formula for finding a woman's height using the femur

Use the formula for women's height based on the femur length: \[ h(r) = 61.41 + 2.32r \]

08

Substitute the femur length into the formula

Substitute the given femur length of 50 cm into the formula. \[ h(50) = 61.41 + 2.32 \times 50 \]

09

Calculate the height for the given femur length

Perform the calculation: \[ h(50) = 61.41 + 116 = 177.41 \] The height of the woman is 177.41 cm.

10

Identify the formula for finding a woman's height using the tibia

Use the formula for women's height based on the tibia length: \[ h(t) = 72.57 + 2.53t \]

11

Substitute the tibia length into the formula

Substitute the given tibia length of 36 cm into the formula. \[ h(36) = 72.57 + 2.53 \times 36 \]

12

Calculate the height for the given tibia length

Perform the calculation: \[ h(36) = 72.57 + 91.08 = 163.65 \] The height of the woman is 163.65 cm.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Equations

Linear equations are mathematical statements that describe a straight-line relationship between two variables. In this exercise, we see linear equations being used to find the height of a person based on the length of their bones. For instance, the formula for a man’s height using the femur length is \[h(r) = 69.09 + 2.24r\]. This equation tells us that the height (\(h\)) increases by 2.24 cm for every 1 cm increase in femur length (\(r\)).Linear equations are straightforward to solve. You can plug in known values to find unknown ones, making them highly useful in fields like forensics or biometrics, where fast and accurate measurements are needed.

Substitution Method

The substitution method is a technique used to solve equations by replacing one variable with an equivalent expression involving another variable. In this exercise, we substitute the given bone lengths into the height formula to find the heights. For example, we use the femur length of 56 cm for a man: \[h(56) = 69.09 + 2.24 \times 56\]This substitution makes the equation only about the height (\(h\)), which simplifies the calculation to finding a single unknown. Simply perform the arithmetic to get the height. This method is widely used in algebra to make complex problems solvable.

Function Evaluation

Function evaluation involves calculating the output of a function given specific input values. For height calculations using bone lengths, we evaluate the function by plugging in the bone length into the formula. For example, to find the height of a man with a tibia length of 40 cm: \[h(40) = 81.69 + 2.39 \times 40\]By performing the arithmetic, we get the height. Function evaluation helps us understand how changing input values (bone lengths) affect the output (height). This concept is essential in many scientific calculations where we have predefined relationships between variables.

Biometric Measurements

Biometric measurements refer to the statistical analysis of people's physical characteristics. These measurements are crucial for identifying individuals and making forensic determinations. In this exercise, the lengths of the femur and tibia are used as biometric measurements to estimate height. The given formulas provide a reliable means of converting bone lengths into height, which is useful for forensic science

• Height estimation from bone lengths allows accurate identification of unknown remains.
• Key in criminal investigations and historical studies.
• Useful in medical fields to understand growth patterns.

Understanding how to use biometric measurements can open doors to various applications in science and law enforcement.