Problem 145 - Algebra and Trigonometry

Question

A bank offers two checking account plans. Plan A has a base service charge of 4.00 dollar per month plus 10¢ per check. Plan B charges a base service charge of $2.00 per month plus 15¢ per check. a. Write models for the total monthly costs for each plan if x checks are written. b. Use a graphing utility to graph the models in the same [0, 50, 10] by [0, 10, 1] viewing rectangle. c. Use the graphs (and the intersection feature) to determine for what number of checks per month plan A will be better than plan B. d. Verify the result of part (c) algebraically by solving an inequality.

Step-by-Step Solution

Answer

a. The models for the costs of the plans are \( A(x) = 4.00 + 0.10x \) and \( B(x) = 2.00 + 0.15x \).\n b. The graphs of the models are numerical interpretations of the models and must be created using a graphing utility. \n c. The intersection on the graph indicates the number of checks that makes both plans cost the same. \n d. To find the number of checks that makes plan A cheaper than plan B, solve the inequality \( A(x)

Step 1: Writing the models

Let's denote x as the number of checks written per month. Plan A charges a flat 4.00 dollars plus 0.10 dollars for each check, resulting in the model \( A(x) = 4.00 + 0.10x \). Plan B charges a base rate of 2.00 dollars plus 0.15 dollars for each check, resulting in the model \( B(x) = 2.00 + 0.15x \).

Step 2: Graphing the models

Next, plot the two models. The x-axis will range from 0 to 50 with intervals of 10, and the y-axis will range from 0 to 10 with intervals of 1. These models should be graphed using a proper graphing utility. Also remember to label the axes properly.

Step 3: Finding the intersection point

Locate the point on the graph where the lines representing the models intersect. This point represents the number of checks per month where both plans have the same cost. Mark this point on your graph.

Step 4: Verifying the result

To find out for what number of checks plan A is more beneficial than plan B, solve the inequality \( A(x)

Key Concepts

Algebraic Modeling

Algebraic modeling is a mathematical method that involves using algebra to create equations or models that represent real-world scenarios. In the context of comparing checking account plans, one must understand how algebra can represent the costs associated with different banking options.

The models for bank plans A and B are created by translating fixed costs and variable costs into algebraic expressions. For Plan A, with a \(4.00 base charge and \)0.10 per check, the total monthly cost is represented by the linear model A(x) = 4.00 + 0.10x where x is the number of checks written. Plan B's model, with a \(2.00 base charge and \)0.15 per check, is B(x) = 2.00 + 0.15x.

Fundamentally, algebraic modeling in this case provides a way to easily compare the costs of the two plans based on the number of checks written and to predict future costs.

Graphing Linear Equations

Graphing linear equations is an essential part of understanding how different equations compare. By graphing the linear models for Plans A and B, you can visualize the relationship between the number of checks written and the total cost for each plan.

To graph the linear equations representing the total monthly costs, you use the horizontal axis (x-axis) for the quantity of checks and the vertical axis (y-axis) for the total cost. Each plan's model results in a straight line where the y-intercept represents the base service charge and the slope of the line indicates the cost per check.

A graphing utility helps to plot these lines and the points of intersection. In our problem, using a [0, 50, 10] by [0, 10, 1] viewing rectangle, you graph A(x) and B(x) for the values of x from 0 to 50 checks. This visual comparison helps expose the number of checks where one plan becomes more cost-effective than the other.

Solving Inequalities

Solving inequalities is a mathematical process used to find the values for which one expression is less than, greater than, equal to, or not equal to another. When comparing the costs of two checking account plans, you'll need to identify where one cost is less than the other.

In this context, you'll solve the inequality A(x) < B(x) to determine the number of checks for which Plan A is cheaper than Plan B. To do so, solve the corresponding equation A(x) = B(x) to determine the break-even point—the number of checks where both plans cost the same.

The solution involves finding the x-value at which both plans intersect on the graph. You then examine the values on either side of this break-even point to establish the interval in which Plan A is more cost-effective. Understanding how to manipulate and solve inequalities is crucial in making informed decisions based on mathematical models.

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Key Concepts

Algebraic Modeling

Algebraic modeling is a mathematical method that involves using algebra to create equations or models that represent real-world scenarios. In the context of comparing checking account plans, one must understand how algebra can represent the costs associated with different banking options.

The models for bank plans A and B are created by translating fixed costs and variable costs into algebraic expressions. For Plan A, with a \(4.00 base charge and \)0.10 per check, the total monthly cost is represented by the linear model A(x) = 4.00 + 0.10x where x is the number of checks written. Plan B's model, with a \(2.00 base charge and \)0.15 per check, is B(x) = 2.00 + 0.15x.

Fundamentally, algebraic modeling in this case provides a way to easily compare the costs of the two plans based on the number of checks written and to predict future costs.

Graphing Linear Equations

Graphing linear equations is an essential part of understanding how different equations compare. By graphing the linear models for Plans A and B, you can visualize the relationship between the number of checks written and the total cost for each plan.

To graph the linear equations representing the total monthly costs, you use the horizontal axis (x-axis) for the quantity of checks and the vertical axis (y-axis) for the total cost. Each plan's model results in a straight line where the y-intercept represents the base service charge and the slope of the line indicates the cost per check.

A graphing utility helps to plot these lines and the points of intersection. In our problem, using a [0, 50, 10] by [0, 10, 1] viewing rectangle, you graph A(x) and B(x) for the values of x from 0 to 50 checks. This visual comparison helps expose the number of checks where one plan becomes more cost-effective than the other.

Solving Inequalities

Solving inequalities is a mathematical process used to find the values for which one expression is less than, greater than, equal to, or not equal to another. When comparing the costs of two checking account plans, you'll need to identify where one cost is less than the other.

In this context, you'll solve the inequality A(x) < B(x) to determine the number of checks for which Plan A is cheaper than Plan B. To do so, solve the corresponding equation A(x) = B(x) to determine the break-even point—the number of checks where both plans cost the same.

The solution involves finding the x-value at which both plans intersect on the graph. You then examine the values on either side of this break-even point to establish the interval in which Plan A is more cost-effective. Understanding how to manipulate and solve inequalities is crucial in making informed decisions based on mathematical models.