Problem 96 - 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 feet in \(x\) miles

Step-by-Step Solution

Answer

The formula is \( g(x) = 5280 \cdot x \). For 10 miles, there are 52,800 feet.

Step 1: Identify the Conversion Factor

To calculate the number of feet in a given number of miles, we need to know the conversion factor. There are 5280 feet in one mile.

Step 2: Write the Function Formula

The function \( g(x) \) calculates the number of feet in \( x \) miles. Using the conversion factor from Step 1, the formula can be written as: \[ g(x) = 5280 \cdot x \]

Step 3: Evaluate the Function at 10 Miles

Now we need to calculate \( g(10) \). Substitute 10 for \( x \) in the function: \[ g(10) = 5280 \cdot 10 \] Perform the multiplication to get: \[ g(10) = 52800 \]

Step 4: Interpret the Result

The result \( g(10) = 52800 \) means that there are 52,800 feet in 10 miles.

Key Concepts

Conversion Factors

Conversion factors are essential in changing units from one measurement to another. In this exercise, the conversion factor needed is between miles and feet.

• One mile is equivalent to 5280 feet.
• This conversion factor allows us to convert the distance given in miles into feet by multiplying by 5280.

For example, if you need to find the number of feet in 5 miles, you simply multiply 5 by 5280. Understanding conversion factors helps in accurately transforming measurements from one unit system to another. It's a fundamental skill not just in algebra but in various real-world applications.

Function Evaluation

Function evaluation is the process where you substitute a given value into a function to determine the output. For the function defined as \( g(x) = 5280 \cdot x \), evaluation is straightforward.
First, identify the input value \( x \). In this exercise, we evaluated the function at 10 miles.

• You replace \( x \) with 10 in the function formula, resulting in \( g(10) = 5280 \times 10 \).
• Performing the multiplication, we find \( g(10) = 52800 \).

Each different value of \( x \) will give you a different result when substituted into the function. Function evaluation helps in providing concrete solutions by plugging in specific inputs into the mathematical representation provided by the function.

Symbolic Representation

Symbolic representation in algebra provides a concise way to express mathematical relationships using symbols and formulas. In this exercise, a function was created to model the relationship between miles and feet.

• The function \( g(x) = 5280 \cdot x \) symbolically represents converting miles into feet.
• Here, \( g(x) \) is the output or the total number of feet, while \( x \) is the number of miles.

Symbolic representation is powerful because it allows for easy manipulation and evaluation of the relationship it describes. By simply substituting different values of \( x \), you can determine the number of feet for any given number of miles. This mathematical shorthand is particularly useful in creating formulas that apply universally to a wide range of situations, enhancing problem-solving efficiency.