Problem 83 - Algebra and Trigonometry

Question

Car Rental cost A car rental company offers two plans for renting a car: Plan A: \(\$ 30\) per day and 10\(c\) per mile Plan B: \(\$ 50\) per day with free unlimited mileage For what range of miles will Plan B save you money?

Step-by-Step Solution

Answer

Plan B saves money if you drive more than 200 miles per day.

Step 1: Define the cost equations

Let's define the cost equations for each plan. For Plan A, the cost is \(C_A = 30d + 0.10m\), where \(d\) is the number of days and \(m\) is the number of miles. For Plan B, since mileage is free, the cost is \(C_B = 50d\), where \(d\) is the days rented.

Step 2: Set up the inequality

We want to find when Plan B saves you money compared to Plan A. This means Plan B should cost less: \(C_B < C_A\). Substitute the formulas: \(50d < 30d + 0.10m\).

Step 3: Solve for the number of miles, m

To find \(m\), subtract \(30d\) from both sides: \(20d < 0.10m\). Then, divide by 0.10: \(m > \frac{20d}{0.10}\). Simplifying this gives \(m > 200d\).

Step 4: Interpret the inequality

The inequality \(m > 200d\) means that Plan B saves money if you drive more than 200 miles per day rented.

Key Concepts

Cost Equations

In the context of renting a car, understanding the cost structure of different plans can guide you to make economical choices. For the two plans under discussion, each has its unique cost equation. In Plan A, the cost is determined by both the number of days and the mileage driven. Specifically, you would pay \( C_A = 30d + 0.10m \), where \( d \) is the number of days and \( m \) the miles driven. This equation shows the daily cost as a fixed amount with additional costs scaling linearly with mileage.
Plan B, on the other hand, is much simpler. It doesn't depend on mileage, only on time: \( C_B = 50d \). No matter how far you drive, the cost per day is fixed because the mileage is free. This straightforward cost equation reflects a higher daily rental fee but potentially fewer costs in terms of distance driven.
When approaching car rental problems, identifying and understanding these cost equations is critical. It allows you to compare different rental scenarios and assess which option better fits your driving needs.

Inequality Solving

Inequality solving plays a crucial role in determining under which conditions one car rental plan is more advantageous than another. Once the cost equations are established, the next step is setting up and solving an inequality to find threshold points. In our scenario, we wish to determine when Plan B saves more money, so we begin by setting up the inequality \( C_B < C_A \), substituting the cost equations to represent this as \( 50d < 30d + 0.10m \).
To solve for the number of miles \( m \), isolate \( m \) by first bringing like terms together: subtract \( 30d \) from both sides resulting in \( 20d < 0.10m \). The next step involves dividing both sides by 0.10 to further isolate \( m \), thus giving \( m > \frac{20d}{0.10} \). Simplifying this inequality we arrive at \( m > 200d \).
This simple inequality solution practice provides a clear boundary for mileage where one plan is better than the other. It emphasizes the importance of comparing not just the costs, but also understanding how factors like mileage impact overall costs across different plans.

Mileage Calculation

Mileage calculation is the linchpin that determines the more cost-effective plan in car rental equations. When mileage impacts cost, as with Plan A, calculating its effect accurately is key. In this problem, the solution to the inequality \( m > 200d \) clarifies how far you need to drive before Plan B becomes cost-effective. Simply put, Plan B saves money if you travel more than 200 miles for each day you rent the car.
This calculation simplifies decision-making for renters. Before renting, anticipate the number of days and miles likely to be covered. If a trip's expected mileage exceeds the threshold indicated in your calculations, Plan B would be advantageous. On the other hand, for shorter distances below this boundary, Plan A might be more economical.
By understanding the significance of mileage calculation, renters can better plan their trips and manage costs, turning a potentially complex decision into a straightforward comparison.