Problem 76 - Algebra A Combined Function

Question

Forensic scientists use the following functions to find the height of a woman if they are given the length of her femur bone \((f)\) or her tibia bone \((t)\) in centimeters. \(H(f)=2.59 f+47.24\) \(H(t)=2.72 t+61.28\) Use these functions to answer Exercises 75 and 76 Find the height of a woman whose tibia measures 35 centimeters.

Step-by-Step Solution

Answer

The woman's height is 156.48 cm.

Step 1: Identify the Problem

We are given the function for height based on the tibia length, which is \( H(t)=2.72 t+61.28 \). We need to find the height of a woman whose tibia is 35 cm long.

Step 2: Substitute the Given Tibia Length

Substitute \( t = 35 \) into the equation \( H(t) = 2.72 t + 61.28 \).

Step 3: Calculate the Product

Calculate \( 2.72 \times 35 \). This gives you 95.2.

Step 4: Add the Constant

Add the constant 61.28 to the result from the previous step: \( 95.2 + 61.28 \).

Step 5: Compute the Final Height

Calculate \( 95.2 + 61.28 \) to obtain the final height. This results in 156.48.

Key Concepts

Linear Equations

Linear equations represent relationships where there is a constant rate of change between two variables; in mathematical terms, this relationship is depicted as a straight line when graphed. A linear equation in one variable has the format: \( y = mx + b \), where:

• \( y \) is the dependent variable or the output of the function.
• \( x \) is the independent variable or input.
• \( m \) represents the slope, indicating how much \( y \) changes for a one-unit change in \( x \).
• \( b \) is the y-intercept, which is the value of \( y \) when \( x = 0 \).

In our forensic example, the heights of women are calculated based on their bone measurements using linear equations. The tibia bone height function is \( H(t) = 2.72t + 61.28 \), where \( t \) represents the tibia bone length in centimeters. Here, 2.72 is the rate at which height increases for each additional centimeter of tibia bone. The constant 61.28 reflects the baseline height corresponding to zero tibia length, purely for functional purposes.

Problem-Solving Steps

Effective problem-solving requires a systematic approach to ensure that no aspect of the problem is overlooked. Let's break down the steps:

• Identify the problem: Understand what the function is and what problem needs to be solved. In this case, the functional relationship between tibia length and height.
• Substitute known values: Next, input any given or known values into the equation or function. Here, substituting \( t = 35 \) into the equation yields \( H(t) = 2.72 \times 35 + 61.28 \).
• Perform calculations: Compute any necessary arithmetic involving multiplication, division, addition, or subtraction. For example, calculate \( 2.72 \times 35 \) and then add 61.28 to find the height.
• Analyze results: Finally, interpret the calculated result in the context of the problem to ensure it makes sense and addresses the given question.

Substitution Method

The substitution method is a technique used in algebra to replace variables with known values, simplifying the process of solving equations. This approach is particularly beneficial when dealing with equations where one variable's value is already known.In our example, the substitution method involved using the known tibia bone length (\( t = 35 \)) in the height function \( H(t) = 2.72t + 61.28 \). Here's how it works:

• Identify the variable that has a given value — in our case, \( t = 35 \).
• Substitute this value into the equation to replace the variable \( t \), resulting in a specific numeric expression: \( H(35) = 2.72 \times 35 + 61.28 \).
• Solve the equation by carrying out the arithmetic operations, hence deriving a precise output: the calculated height of 156.48 cm for the woman.

This method is a straightforward yet powerful algebraic tool, assisting in translating abstract equations into tangible results.