Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. The relationship between the height \(H\) of an adult female and the length \(x\) of her tibia, in centimeters, is estimated by the linear model \(H(x)=2.90 x+61.53 .\) If incomplete skeletal remains of an adult female include a tibia measuring 30.9 centimeters, estimate the height of the female. Round to the nearest tenth.

Short answer

The estimated height of the female is 151.1 cm.

Step-by-step solution

Understand the Given Linear Model

We are given a linear model that estimates the height of an adult female based on the length of her tibia: \( H(x) = 2.90x + 61.53 \). In this model, \( x \) is the length of the tibia in centimeters and \( H(x) \) is the height in centimeters.

Substitute the Tibia Length into the Model

We are given that the tibia length (\( x \)) is 30.9 centimeters. Substitute \( x = 30.9 \) into the linear model: \( H(30.9) = 2.90(30.9) + 61.53 \).

Perform the Multiplication

First, calculate the multiplication part of the model: \( 2.90 \times 30.9 \). \( 2.90 \times 30.9 = 89.61 \).

Add the Constant Term

Next, add the constant term from the model to the product we calculated: \( 89.61 + 61.53 \). \( 89.61 + 61.53 = 151.14 \).

Round to the Nearest Tenth

Finally, round 151.14 to the nearest tenth to get the estimated height. \( 151.14 \) rounded to the nearest tenth is \( 151.1 \).

Key concepts

linear equations

Linear equations are the backbone of many mathematical models. A linear equation can be written in the form:

\[ y = mx + c \]
here,

• \(y\): dependent variable
• \(x\): independent variable
• \(m\): slope (rate of change)
• \(c\): y-intercept (value of y when x is zero)

In the given exercise, the linear model is written as

\[ H(x) = 2.90x + 61.53.\]
Here, \(H(x)\) is the estimated height based on the tibia measurement \(x\). Understanding and manipulating such equations is essential for solving real-world problems like height estimation.

tibia bone measurement

The tibia is a key bone in the human leg, extending from the knee to the ankle. It is often measured to estimate an individual's height.

Anthropologists and forensic scientists use tibia measurements in their analysis because the length of the tibia correlates strongly with overall height.

In this exercise, the tibia length is given as 30.9 cm, which we then use in the linear model to estimate the height of an adult female. Such measurements are crucial in archaeological and forensic science for reconstructing the physical attributes of individuals from skeletal remains.

height estimation

Height estimation involves predicting an individual's height from various body measurements. One common method uses mathematical models like linear equations.

In this particular exercise, the linear equation provided is used to estimate height from tibia length:

\[ H(x) = 2.90x + 61.53 \]
By substituting the tibia length (30.9 cm) into the equation, we calculate the height:

• First, substitute the value: \[ H(30.9) = 2.90 \times 30.9 + 61.53 \]
• Then perform the calculation: \[ 2.90 \times 30.9 = 89.61 \]
• Finally, add the constant term: \[ 89.61 + 61.53 = 151.14 \]
• Round to the nearest tenth: 151.1 cm.

This gives us an estimated height of 151.1 cm.

substitution in algebra

Substitution is a fundamental skill in algebra used to simplify and solve equations. It involves replacing a variable with its numerical value.

In our exercise, we substitute the given tibia length (30.9 cm) into the linear equation:

\[ H(x) = 2.90 \times 30.9 + 61.53 \]
By doing this, we turn the algebraic equation into a simple arithmetic problem.

This step-by-step substitution and subsequent calculations help to estimate the height:

• Perform multiplication first: 2.90 times 30.9 equals 89.61.
• Then add the constant term: 61.53, resulting in 151.14.
• Finally, round off: Height becomes 151.1 cm.

This clear and methodical process ensures accuracy and ease of understanding.