Problem 93 - 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 dollars in \(x\) quarters

Step-by-Step Solution

Answer

The function is \( g(x) = 0.25x \). Evaluating \( g(10) \) gives $2.50, meaning 10 quarters equal $2.50.

Step 1: Understand the Problem

We need to write a function that calculates the amount of money in dollars given a number of quarters. Quarters are worth $0.25 each.

Step 2: Define the Function

The function \( g(x) \) will calculate the total dollar amount from \( x \) quarters. Since each quarter is worth $0.25, we can express this as \( g(x) = 0.25x \).

Step 3: Evaluate the Function

To find \( g(10) \), substitute 10 for \( x \) in the function: \[ g(10) = 0.25 \times 10. \]

Step 4: Calculate the Result

Perform the multiplication to find the result: \[ g(10) = 2.5. \] So, 10 quarters amount to $2.50.

Key Concepts

Symbolic Representation

Understanding symbolic representation in algebra is crucial, as it allows us to describe real-world situations using mathematical expressions. In this exercise, we are tasked with representing the number of dollars obtained from quarters using a function. Each quarter is worth $0.25, which is a fixed value. We assign this value to a variable, usually denoted by a symbol like \(x\), which in this context represents the number of quarters.
The symbolic representation we use is a function named \(g(x)\). It links the value of \(x\) quarters directly to their monetary value in dollars. The function is expressed as \(g(x) = 0.25x\). This communicates that for any number \(x\) of quarters, the dollar amount is 0.25 times \(x\). This concise formula captures the relationship between quarters and their total dollar value.

Function Evaluation

Once the function is defined, the next step is to evaluate it for specific inputs—this is known as function evaluation. Here, we need to find the value of \(g(10)\). This involves a simple substitution process:

• Take the defined function \(g(x) = 0.25x\).
• Replace \(x\) with 10, because we have 10 quarters: \(g(10) = 0.25 \times 10\).

This substitution is crucial because it allows us to calculate the specific dollar value associated with 10 quarters. The function evaluation is simply the act of plugging in a number for \(x\) and performing the arithmetic operation. In this case, multiplying 0.25 by 10 gives us 2.5. This means that 10 quarters are equal to $2.50.

Interpreting Results

Interpreting the results of a function evaluation is an important aspect of solving algebraic problems. After calculating \(g(10) = 2.5\), we interpret this numerically and contextually. The value 2.5 refers to $2.50, which is the amount of money you have if you have 10 quarters.
This interpretation requires connecting the mathematical result back to the original problem statement. Here, we were asked to determine the dollar amount of a certain number of quarters. The interpretation step confirms that the symbolic representation and evaluation were performed correctly, and the numerical result makes sense within the real-world context. Understanding this strengthens our ability to link mathematical formulas to practical applications.