Problem 47 - College Algebra

Question

A phone company has a monthly cellular plan where a customer pays a flat monthly fee and then a certain amount of money per minute used on the phone. If a customer uses 410 minutes, the monthly cost will be \(\$ 71.50\) . If the customer uses 720 minutes, the monthly cost will be \(\$ 118\) . a. Find a linear equation for the monthly cost of the cell plan as a function of \(x,\) the number of monthly minutes used. b. Interpret the slope and \(y\) -intercept of the equation. c. Use your equation to find the total monthly cost if 687 minutes are used.

Step-by-Step Solution

Answer

The linear equation is \( C(x) = 0.15x + 10 \). The cost for 687 minutes is \$103.05.

Step 1: Identify Variables and Information

We need to find a linear relationship between the monthly cost and minutes used. Let's denote the number of minutes as \( x \), and the total cost as \( C(x) \). We have two data points: (410 minutes, \(71.50) and (720 minutes, \)118).

Step 2: Calculate the Slope of the Line

The formula for the slope \( m \) is given by \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Using our points \((x_1, y_1) = (410, 71.50)\) and \((x_2, y_2) = (720, 118)\), we find:\[m = \frac{118 - 71.50}{720 - 410} = \frac{46.50}{310} \approx 0.15\]

Step 3: Write the Linear Equation

The equation of a line in slope-intercept form is \( C(x) = mx + b \). We already calculated \( m = 0.15 \). Now we can use one of the points to solve for \( b \). Using the point (410, 71.50):\[71.50 = 0.15 \times 410 + b\]\[b = 71.50 - 61.50 = 10\]Thus, the equation is \( C(x) = 0.15x + 10 \).

Step 4: Interpret the Slope and Intercept

The slope \( m = 0.15 \) represents the cost per minute used. In other words, each additional minute costs \( \$0.15 \). The \( y \)-intercept \( b = 10 \) represents the flat monthly fee charged regardless of minutes used.

Step 5: Calculate Cost for 687 Minutes

Using the equation \( C(x) = 0.15x + 10 \), substitute \( x = 687 \):\[C(687) = 0.15 \times 687 + 10 = 103.05\]Thus, the total monthly cost for 687 minutes is \$103.05.

Key Concepts

Understanding Slope in Linear Equations

The slope of a line in a linear equation is a crucial concept that helps you understand how one variable affects another. In our exercise, the slope is determined by the rate at which the cost changes with respect to the number of minutes used on the phone. We calculated the slope (\( m = 0.15 \)) using the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
• Where \( (x_1, y_1) = (410, 71.50) \) and \( (x_2, y_2) = (720, 118) \)
• So, \( m = \frac{118 - 71.50}{720 - 410} = \frac{46.50}{310} \approx 0.15 \)

The slope of \( 0.15 \) indicates that every additional minute costs \( \$0.15 \). This is a great example of how a slope represents a rate of change between two variables, showing how much one variable will increase based on the increase of another.

Decoding the Y-Intercept

The y-intercept in a linear equation is where the line crosses the y-axis, representing the starting point of your equation when the other variable is zero. In this scenario, the y-intercept is the flat fee of the cellular plan before any minutes are used.We calculated the y-intercept \( ( b = 10 ) \) using the slope-intercept form of the equation:

• Using the equation \( C(x) = 0.15x + b \)
• Substituting in one of our points \((410, 71.50)\)
• Solved to find \( b: 71.50 = 0.15 \times 410 + b \)
• This gives \( b = 10 \)

The y-intercept value of \( 10 \) denotes the fixed monthly cost, irrespective of the minutes used. It sets the baseline cost that is then adjusted based on usage.

Linear Functions and Their Application

A linear function is a type of function that creates a straight line when graphed. It's an essential concept for understanding relationships between variables. In this exercise, the monthly cost is expressed as a linear function of the number of minutes used (\( C(x) = 0.15x + 10 \)).Linear functions have two main components:

• The slope \( m \), which shows the rate of change
• The y-intercept \( b \), which accounts for the starting value when all other inputs are zero

These functions are widely used in many real-world applications, such as determining costs, predicting profits, or analyzing data trends. By defining cost as a linear function of minutes, we can easily estimate expenses based on usage.

Cost Analysis Using Linear Equations

Cost analysis involves understanding how different variables influence the cost in a given scenario. Using linear equations is a straightforward method to model this relationship precisely. In our example, the cost equation \( C(x) = 0.15x + 10 \) is applied to analyze and predict the total cost for a given number of minutes.Here's how cost analysis works in this context:

• Identify fixed costs represented by the y-intercept \( 10 \)
• Recognize variable costs indicated by the slope \( 0.15 \) per additional minute
• Predict costs using the function, such as calculating \( C(687) \) minutes to find \( C(687) = 0.15 \times 687 + 10 = 103.05 \)

Cost analysis helps in better financial planning by offering insights into fixed and variable expenses, enabling customers to make informed decisions about their phone usage.