Problem 24 - College Algebra

Question

For the following exercises, use this scenario: A truck rental agency offers two kinds of plans. Plan A charges \(\$ 75 / \mathrm{wk}\) plus \(\$ .10 / \mathrm{mi}\) driven. Plan B charges \(\$ 100 / \mathrm{wk}\) plus \(\$ .05 \mathrm{mi}\) driven. Write the model equation for the cost of renting a truck with plan A.

Step-by-Step Solution

Answer

The model equation for Plan A is \(C_A = 75 + 0.10m\).

Step 1: Identify Fixed and Variable Costs for Plan A

First, determine the fixed weekly cost and the variable cost per mile for Plan A. The weekly fixed cost is $75. The variable cost is $0.10 per mile driven.

Step 2: Define Variables

Define the variable \(C_A\) as the total cost using Plan A, and \(m\) as the number of miles driven.

Step 3: Construct the Model Equation

Using the information from Steps 1 and 2, create an equation to model the total cost for Plan A. The equation is: \(C_A = 75 + 0.10m\), where \(C_A\) is the total cost and \(m\) is the total miles driven.

Key Concepts

Equation Formation

To understand equation formation, consider a scenario where you want to calculate the total cost of renting a service, such as a truck. You need a structured way to express this calculation. This is where forming equations becomes essential. The general idea is to represent unknown quantities with variables, which can then be assembled into an equation using numerical values and functional relationships.

In our truck rental example, we want to find a formula for the cost based on usage. You start this process by writing down what you know:

• A fixed cost is the set fee you always pay, regardless of additional usage. Here, it's the weekly rental fee of \(75.
• A variable cost, however, depends on your usage level, such as driving more miles. This cost is \)0.10 per mile in Plan A.
• Finally, define meaningful variables for unknown quantities: let \(C_A\) represent the total cost and \(m\) represent the miles driven.

Thus, the equation for Plan A becomes \(C_A = 75 + 0.10m\). It shows that the total cost \(C_A\) results from adding the fixed fee to the product of miles driven and cost per mile. This equation is your algebraic model, streamlining cost calculations into a clear and organized form.

Fixed and Variable Costs

Understanding fixed and variable costs is critical when analyzing offers like rental plans. Fixed costs are static and remain constant regardless of conditions. In the truck rental scenario, the fixed cost is the weekly charge of $75 for Plan A.

Variable costs, on the other side, change with usage. They scale based on the level of activity or consumption. For Plan A, the variable cost is the $0.10 per mile driven. This type of cost allows customers to estimate their final charge by considering their specific use case.

Fixed costs provide stability, allowing consumers to easily picture a baseline expense. Variable costs introduce flexibility, reflecting real usage, encouraging efficient and mindful consumption. Always consider both when determining which rental or service plan best suits your needs.

Cost Analysis

Cost analysis evaluates financial implications and aids in decision-making. It's crucial to choose the best financial strategy, especially when multiple options are available, as in the truck rental plans.

To perform effective cost analysis, begin by comparing the overall expenses of each plan. Identify which costs apply in your specific situation. Consider not just the rates but also expected usage, like estimated miles driven for our truck plans.

With the equations formed (e.g., \(C_A = 75 + 0.10m\) for Plan A), plug in different mileage values to compare which plan might be cheaper at various levels of use. This analysis quantifies the repercussions of each additional mile driven or each fixed weekly rental.

Ultimately, cost analysis empowers individuals to make informed choices based on projected needs and budgetary limits, ensuring that the selected plan aligns with financial goals and expected usage patterns.