Problem 145 - College Algebra Essentials

Question

A bank offers two checking account plans. Plan A has a base service charge of \(\$ 4.00\) per month plus \(10 \not c\) per check. Plan B charges a base service charge of \(\$ 2.00\) per month plus \(15 \not c\) 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 are \(C_A = 4 + 0.10x\), \(C_B = 2 + 0.15x\). \nb. The graphs can be plotted within the given rectangle domain using a graphing utility. c. The intersection point on the graph gives the number of checks where the cost for both plans is the same. d. Solving the inequality \(C_A < C_B\) algebraically for \(x\) verifies the number of checks for which Plan A is better than Plan B.

Step 1: Building the Models

We can write two equations to represent monthly costs (C) in dollars for each plan. For Plan A, the equation would be \(C_A = 4 + 0.10x\), and for plan B, it would be \(C_B = 2 + 0.15x\). Here, \(x\) is the number of checks written each month.

Step 2: Graph the Models

You use a graphing utility (like a graphing calculator or a computer software), plot both \(C_A\) and \(C_B\) in the given \([0,50,10]\) by \([0,10,1]\) viewing rectangle. This can show how the cost changes as the number of checks changes.

Step 3: Determining the Intersection

On the graph from step 2, find where the two equations intersect. This intersection point represents the number of checks at which two plans cost the same. The x-coordinate of the point is what you're looking for.

Step 4: Solve an Inequality

To verify the result of part c algebraically, set up an inequality where \(C_A < C_B\) (it represents when plan A is cheaper than plan B) and solve for \(x\). This gives the number of checks, beyond which Plan A is more economical.

Key Concepts

Modeling

When we talk about modeling in mathematics, we're essentially creating a representation of a real-world situation using mathematical expressions. In this exercise, the task is to model the monthly costs of two bank account plans. Modeling allows us to use known variables and conditions to create equations to predict outcomes or to understand the relationship between different quantities.

Here's how it works for this problem: we're given two plans with different base charges and per check costs. Plan A has a base service charge of \(4 and \)0.10 per check, while Plan B has a \(2 base and \)0.15 per check. The goal is to come up with expressions (or models) for each plan based on the number of checks, denoted by 'x'. This process involves crafting linear equations, which are foundational in modeling real-world financial situations.

To do this, the costs can be described with these equations:

• Plan A: \(C_A = 4 + 0.10x\)
• Plan B: \(C_B = 2 + 0.15x\)

These models now allow us to calculate and compare the total monthly costs based on any given number of checks.

Graphing Utilities

Graphing utilities are tools that help us visualize mathematical equations. They play an essential role in understanding how variables interact in various mathematical models. In this exercise, we'll use graphing utilities to visualize the cost models we've created for the two plans.

By plotting the two cost equations on a graph, we put Plan A and Plan B in a 'viewing rectangle', which is a defined space on the graph where we can clearly see how each plan behaves as the number of checks changes. For this exercise, the viewing rectangle is set between the intervals ":[0,50,10]" for the x-axis (number of checks) and ":[0,10,1]" for the y-axis (cost in dollars). This means we're observing these costs in a reasonable check-writing range.

• Plan A appears as a line starting from an intercept of $4 and with a lesser slope compared to Plan B.
• Plan B starts at $2 but with a steeper slope due to the higher per-check charge.

These visual cues from graphing utilities give us immediate understanding of how costs accumulate under both plans, aiding in strategic decision-making.

Intersection

In mathematics, the intersection of two graphs is where they meet – that is, the point where both equations yield the same value. This is a crucial concept, especially when comparing different plans or options like in this bank account example.

After graphing the equations for Plan A and Plan B, the intersection point is identified. This is done by looking at where both lines cross each other on the graph. The coordinates of this point (both x and y) tell us:

• The x-coordinate reveals the number of checks at which both plans cost the same.
• The y-coordinate is the cost at that equivalence point.

Finding the intersection not only solves algebraic equations visually but also aids us in understanding over what range of checks Plan A might become more beneficial. Specifically, if you need to choose which plan to go for beyond this equivalence point, checking how each one progresses from this meeting point is key in decision-making.

Inequalities

Inequalities, much like equations, involve comparison between expressions. However, instead of showing equality (\(=\)), they show a range of values or conditions (\(<\) or \(>\)). In this exercise, we use inequalities to determine which bank plan is more economical under given conditions.

For the exercise, you want to ascertain when Plan A becomes a cheaper option than Plan B. To do this, we set up an inequality:

• \(C_A < C_B\), or explicitly \(4 + 0.10x < 2 + 0.15x\)

Solving this involves typical algebraic steps:

• First, subtract 0.10x from both sides: \(4 < 2 + 0.05x\)
• Then subtract 2 from both sides: \(2 < 0.05x\)
• Finally, divide by 0.05 to isolate x: \(x > 40\)

Hence, Plan A becomes more economical for any number of checks greater than 40 per month. Inequalities like this help quantify at which point one option outperforms another in real-life situations.