Problem 27 - 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. If Tim knows he has to travel 300 \(\mathrm{mi}\) , which plan should he choose?

Step-by-Step Solution

Answer

Tim should choose Plan A, as it costs less for 300 miles.

Step 1: Calculate Total Cost for Plan A

Start by calculating the total cost for Plan A using the formula: \( \text{Total Cost}_{A} = \\(75 + (0.10 \times \text{miles}) \). Substitute 300 for miles to get \( \text{Total Cost}_{A} = \\)75 + (0.10 \times 300) = \\(75 + \\)30 = \$105 \).

Step 2: Calculate Total Cost for Plan B

Next, calculate the total cost for Plan B using the formula: \( \text{Total Cost}_{B} = \\(100 + (0.05 \times \text{miles}) \). Substitute 300 for miles to get \( \text{Total Cost}_{B} = \\)100 + (0.05 \times 300) = \\(100 + \\)15 = \$115 \).

Step 3: Compare the Total Costs

Now that you have the total costs for both plans, compare them. Plan A costs \\(105, while Plan B costs \\)115. As Plan A is cheaper, Tim should choose Plan A.

Key Concepts

Algebraic Problem Solving

Algebraic problem solving is like a puzzle. It involves using mathematical equations and expressions to find unknown values or make comparisons, like in our truck rental scenario. Here, algebra helps us decide which plan is more cost-effective for Tim. We use equations to represent the total cost for each plan.
Each plan has a fixed weekly charge and a variable charge that depends on the number of miles driven. We express these charges as formulas:

• Plan A: \( 75 + 0.10 \times ext{miles} \)
• Plan B: \( 100 + 0.05 \times ext{miles} \)

Substituting numbers into these equations can help Tim compare the costs. For example, by substituting 300 miles, we calculate the total costs for Plan A and B. This step-by-step calculation simplifies the decision-making process, allowing Tim to determine the cheaper option.

Linear Equations

Linear equations are powerful mathematical tools. They are equations that form a straight line when graphed. In our truck rental scenario, they help to model the cost associated with each rental plan.
A linear equation typically takes the form \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept. Here, the cost equations for the plans fit this form, with 'miles' acting as the variable \( x \):

• Plan A: \( y = 0.10 \times ext{miles} + 75 \)
• Plan B: \( y = 0.05 \times ext{miles} + 100 \)

The coefficient of the 'miles' term (\( m \)) represents the rate at which the cost increases per mile driven, while the constant term (\( b \)) represents the base weekly charge. Solving these equations helps us find out at which mileage both plans incur the same cost, or lets us analyze costs for various distances, such as the 300 miles in Tim's scenario.

Mathematical Reasoning

Mathematical reasoning is the logic used to solve problems and make decisions. It involves critical thinking and a structured approach to analyzing situations - like deciding between Plan A and B in our problem.
Tim needs to understand how the charges in each plan impact the overall cost. Mathematical reasoning helps him move from raw data (cost per week and mile) to conclusions about which rental plan is cheaper for a specific mileage. We use calculations to form these conclusions logically.

• Calculate each plan's cost for the given miles.
• Compare these costs directly.

This type of reasoning not only helps in determining the outcome in this scenario but also instills a rationale useful in many real-life situations. By systematically comparing the cost of each plan, Tim can make an evidence-based decision, ensuring his choice is economically beneficial.