Problem 27 - College Algebra with Corequisite Support

Question

For the following exercises, use this scenario: A truck rental agency offers two kinds of plans. Plan A charges $$\$ 75 / w$$ k plus $$\$.10 / \mathrm{mi}$$ driven. Plan \(\mathrm{B}\) 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.

Step 1: Define the Cost Functions

Let's define the cost functions for each plan. For Plan A, the cost \( C_A \) is given by \( C_A = 75 + 0.10x \), where \( x \) is the number of miles driven. For Plan B, the cost \( C_B \) is given by \( C_B = 100 + 0.05x \).

Step 2: Substitute the Miles Driven

We know Tim has to drive 300 miles. Substitute \( x = 300 \) into each cost function. For Plan A, \( C_A = 75 + 0.10(300) \). For Plan B, \( C_B = 100 + 0.05(300) \).

Step 3: Calculate the Cost for Plan A

Calculate the cost for Plan A by solving \( C_A = 75 + 0.10 \times 300 = 75 + 30 = 105 \).

Step 4: Calculate the Cost for Plan B

Calculate the cost for Plan B by solving \( C_B = 100 + 0.05 \times 300 = 100 + 15 = 115 \).

Step 5: Compare the Costs

Compare the costs obtained: \( C_A = 105 \) and \( C_B = 115 \). Since \( C_A < C_B \), Plan A is cheaper than Plan B.

Key Concepts

Linear Equations

Linear equations are mathematical expressions where every term is either a constant or involves only one variable raised to the power of one. In our scenario, we are given two cost functions linked to truck rental plans: one for Plan A and one for Plan B. These are linear because they follow the form of a line, expressed generally as \( y = mx + c \). Here, \( y \) represents the cost, \( x \) is the variable miles driven, \( m \) is the cost per mile, and \( c \) is the weekly rental fee.

Understanding linear equations is essential as they help us predict costs given different values of mileage. For Plan A, the equation is \( C_A = 75 + 0.10x \), and for Plan B, it is \( C_B = 100 + 0.05x \). By substituting different values for \( x \), you can easily calculate and compare the costs for varying distances. This property of linear equations makes them very useful for cost function analysis, allowing users to quickly determine the impact of each additional mile on the total cost.

Cost Comparison

When faced with multiple pricing options, cost comparison becomes crucial to make an informed choice. In the truck rental problem, we are tasked with comparing two plans to find out which is more economical over a predetermined distance of 300 miles.

Here are some steps to guide you in cost comparison:

• First, identify the cost structure of each plan, including fixed and variable costs.
• Substitute the known number of miles into the cost equations: \( C_A = 75 + 0.10(300) \) and \( C_B = 100 + 0.05(300) \).
• Calculate the total cost for each plan to ensure accuracy.
• Finally, compare the outputs: for our given scenario, \( C_A = 105 \) and \( C_B = 115 \).

Cost comparison helps you save money by revealing the most affordable option based on your specific criteria—here, the number of miles to be driven.

Decision Making in Mathematics

Mathematics provides a structured approach to decision making, guiding us through situations that involve selecting between alternatives based on calculated outcomes. In scenarios like choosing a truck rental plan, mathematical analysis gives clarity and direction.

Here’s how you can apply mathematical decision making to choose a plan:

• Clearly define the parameters involved—in this case, the number of miles and the associated costs.
• Create a model (cost functions) to evaluate each option accurately.
• Calculate the figures and make comparisons to see which results meet your needs best.
• Choose the option that offers the best balance of cost and benefit.

By using mathematics in decision making, you ensure that you base your choices on solid grounds, minimizing risks and maximizing benefits, which is demonstrated through preferring Plan A over Plan B in our exercise, given it offers a lower cost for the 300-mile journey.