Question

A forensic specialist can estimate the height of a person from the lengths of their bones. One function relates the length, \(x\) of the femur to the height, \(y,\) of the person, both in centimetres. For a male: \(y=2.32 x+65.53\) For a female: \(y=2.47 x+54.13\) a) Determine the height of a male and of a female with a femur length of \(45.47 \mathrm{cm} .\) b) Use inverse functions to determine the femur length of i) a male whose height is \(187.9 \mathrm{cm}\) ii) a female whose height is \(175.26 \mathrm{cm}\)

Short answer

Male: 236.32 cm; Female: 166.45 cm. Male femur: 52.76 cm; Female femur: 49.04 cm.

Step-by-step solution

Determine the height of a male with a femur length of 45.47 cm

Plug the femur length into the formula for males: \[ y = 2.32x + 65.53 \] Substitute \( x = 45.47 \): \[ y = 2.32(45.47) + 65.53 \] Calculate the final value: \[ y = 2.32 \times 45.47 + 65.53 = 170.7904 + 65.53 = 236.3204 \text{ cm} \]

Determine the height of a female with a femur length of 45.47 cm

Plug the femur length into the formula for females: \[ y = 2.47x + 54.13 \] Substitute \( x = 45.47 \): \[ y = 2.47(45.47) + 54.13 \] Calculate the final value: \[ y = 2.47 \times 45.47 + 54.13 = 112.3169 + 54.13 = 166.4469 \text{ cm} \]

Use the inverse function to determine the femur length of a male whose height is 187.9 cm

For males, the formula is: \[ y = 2.32x + 65.53 \] Use the inverse function to solve for \( x \): \[ x = \frac{y - 65.53}{2.32} \] Substitute \( y = 187.9 \): \[ x = \frac{187.9 - 65.53}{2.32} \] Calculate the final value: \[ x = \frac{122.37}{2.32} = 52.75862 \text{ cm} \]

Use the inverse function to determine the femur length of a female whose height is 175.26 cm

For females, the formula is: \[ y = 2.47x + 54.13 \] Use the inverse function to solve for \( x \): \[ x = \frac{y - 54.13}{2.47} \] Substitute \( y = 175.26 \): \[ x = \frac{175.26 - 54.13}{2.47} \] Calculate the final value: \[ x = \frac{121.13}{2.47} = 49.03765 \text{ cm} \]

Key concepts

inverse functions

Inverse functions are a way to reverse the process of a given function. If you have a function that describes how one variable relates to another, its inverse function lets you find the original variable given the second one.
In our problem, we have functions relating femur length to height for males and females. To find the femur length when given a specific height, we use the inverse function:
For males:
\( y = 2.32x + 65.53 \, \rightarrow \, x = \frac{y - 65.53}{2.32} \)
For females:
\( y = 2.47x + 54.13 \, \rightarrow \, x = \frac{y - 54.13}{2.47} \)
The inverse function essentially 'undoes' the original function by swapping the roles of the variables.

linear equations

Linear equations are equations that create straight lines when graphed. They have the general form:
\( y = mx + c \)
where:

• \( m \) is the slope (how steep the line is)
• \( c \) is the y-intercept (where the line crosses the y-axis)

In our problem, the formulas relating femur length to height are linear equations:
For a male:
\( y = 2.32x + 65.53 \)
For a female:
\( y = 2.47x + 54.13 \)
Both of these equations show a direct relationship. As the length of the femur increases, the height also increases. The numbers 2.32 and 2.47 are the slopes, and 65.53 and 54.13 are the y-intercepts.

applied mathematics

Applied mathematics involves using mathematical methods and formulas to solve real-world problems. In our exercise, we are applying mathematical concepts to forensics to predict human height based on bone length.
This involves:

• Understanding relationships between variables (femur length and height)
• Using algebra to manipulate these relationships
• Applying inverse functions to solve for a specific variable

Applied mathematics is essential in many fields such as engineering, physics, economics, and, as we see here, forensics.

height prediction

Height prediction using bone length is a practical example of how mathematics can be used in forensics. By using specific linear equations, forensic specialists can estimate the height of an individual from the length of their femur bone.
For example:

• For a male with a femur length of 45.47 cm:
  \( y = 2.32(45.47) + 65.53 = 170.7904 + 65.53 \approx 236.32 \text{ cm} \)
• For a female with a femur length of 45.47 cm:
  \( y = 2.47(45.47) + 54.13 = 112.3169 + 54.13 \approx 166.45 \text{ cm} \)

This method of height estimation is invaluable in situations where only skeletal remains are present. It is a fascinating intersection of biology and mathematics.