Problem 91 - College Algebra with Modeling and Visualization

Question

Write a symbolic representation (formula) for a function \(g\) that calculates the given quantify. Then evaluate \(g(10)\) and interpret the result. The number of inches in \(x\) feet

Step-by-Step Solution

Answer

The function is \( g(x) = 12x \). So, \( g(10) = 120 \) inches.

Step 1: Define the Relationship

The problem asks us to express the relationship between feet and inches. Knowing that 1 foot equals 12 inches, we can establish this basic conversion factor.

Step 2: Write the Function Formula

Using the relationship from Step 1, we write the function formula. If \(x\) represents the number of feet, then \(g(x) = 12x\) will convert \(x\) feet into inches.

Step 3: Substitute the Given Value

We need to find \(g(10)\). Substitute \(x = 10\) into the formula: \(g(10) = 12 \times 10\).

Step 4: Evaluate the Expression

Calculate the result of the expression from Step 3. \(g(10) = 120\).

Step 5: Interpret the Result

The result, \(g(10) = 120\), means that there are 120 inches in 10 feet.

Key Concepts

Unit Conversion

Understanding unit conversion is essential in mathematics and everyday life. It allows us to convert a quantity expressed in one unit into another equivalent unit. In this exercise, we are converting units of length, specifically from feet to inches. The fundamental conversion factor for this is that 1 foot equals 12 inches. This conversion factor serves as the basis for our function formula.
By multiplying the number of feet by 12, we find the number of inches. Conversion formulas follow a specific unit equation that relates two measurements.
To convert, take the known measurement and multiply it by a conversion factor.

Algebraic Expressions

Algebraic expressions are a fundamental concept in algebra, involving numbers, variables, and operations. In this exercise, we use algebraic expressions to form a relationship between feet and inches. The expression we use is called a function formula.
For example, in this problem, we express the formula as \(g(x) = 12x\). Here,

• \(g(x)\) represents the function that converts feet to inches.
• \(x\) stands for the number of feet.
• The operation \(12x\) represents multiplying the quantity in feet by 12 to get inches.

Understanding algebraic expressions allows for uniformity in expressing mathematical ideas, including unit conversions.

Evaluation of Functions

Evaluating a function involves substituting a particular value for the variable and computing the result. This step applies the function rule to solve problems, as shown by evaluating \(g(10)\) in the exercise.
Here’s how it works:

• Identify the function: \(g(x) = 12x\)
• Substitute the value of the variable: Replace \(x\) with 10, so it becomes \(g(10) = 12 \times 10\)
• Solve the expression: Calculate the product to find the result, \(g(10) = 120\)

This tells us there are 120 inches in 10 feet. Learning to evaluate functions is crucial because it allows mathematicians and students to understand how inputs relate to outputs in mathematical models.