Problem 84 - Algebra and Trigonometry with Analytic Geometry

Question

A consumer is trying to decide whether to purchase car A or car B. Car A costs $$\$ 20,000$$ and has an mpg rating of 30 , and insurance is $$\$ 1000$$ per year. Car B costs $$\$ 24,000$$ and has an mpg rating of 50 , and insurance is $$\$ 1200$$ per year. Assume that the consumer drives 15,000 miles per year and that the price of gas remains constant at \(\$ 3\) per gallon. Based only on these facts, determine how long it will take for the total cost of car B to become less than that of car A.

Step-by-Step Solution

Answer

It will take 10 years for the total cost of Car B to be less than Car A.

Step 1: Determine Annual Fuel Costs

First, calculate the annual fuel cost for each car. We use the formula: \( \text{Annual Fuel Cost} = \left( \frac{\text{Annual Miles Driven}}{\text{Miles Per Gallon}} \right) \times \text{Gas Price} \).For Car A, the annual fuel cost is \( \left( \frac{15000}{30} \right) \times 3 = 1500 \) dollars.For Car B, the annual fuel cost is \( \left( \frac{15000}{50} \right) \times 3 = 900 \) dollars.

Step 2: Determine Total Annual Cost for Each Car

Add the annual fuel cost to the annual insurance cost to compute the total annual cost for each car.For Car A, the total annual cost is: \( 1500 + 1000 = 2500 \) dollars.For Car B, the total annual cost is: \( 900 + 1200 = 2100 \) dollars.

Step 3: Calculate Cost Difference and Initial Price Difference

Determine the difference in the initial price and total annual cost between the two cars.Initial price difference: \( 24000 - 20000 = 4000 \) dollars.Annual cost savings by choosing Car B instead of Car A: \( 2500 - 2100 = 400 \) dollars per year.

Step 4: Solve for the Time to Break Even

To find how many years it will take for Car B's total cost to be less than Car A's, divide the initial price difference by the annual cost savings: \( \frac{4000}{400} = 10 \) years.

Key Concepts

Cost Analysis

When deciding between financial options, understanding all associated costs is crucial. A comprehensive cost analysis involves calculating both upfront and ongoing expenses to understand the total investment over time.
This process helps consumers compare choices effectively, particularly when large-scale expenditures like cars are involved.
For cars, costs aren't limited to the sticker price. They include insurance, fuel, and maintenance, each of which can accumulate significantly over time.

• Initial Purchase Price: The upfront cost of the car itself.
• Annual Fuel Cost: This depends on the car's fuel efficiency and the distance driven.
• Insurance Cost: An ongoing cost that varies based on car type and user profile.

By including all these elements, consumers make informed decisions that reflect the actual cost of ownership over time. Comprehensive cost analysis ensures all expenses are captured, allowing for a more accurate calculation of the total cost.

Annual Fuel Costs

Fuel costs can be a significant portion of a car's total annual expenses. Calculating these costs helps consumers forecast their yearly outlay accurately.
To determine the annual fuel cost, use the formula: \[ \text{Annual Fuel Cost} = \left( \frac{\text{Annual Miles Driven}}{\text{Miles Per Gallon}} \right) \times \text{Gas Price} \]This equation illustrates the impact of a car's fuel efficiency (MPG) on the owner's wallet. Consider two examples:

• **Car A**: With an MPG of 30, driving 15,000 miles a year, the cost at \(3 per gallon would be \)1,500.
• **Car B**: With an MPG of 50 at the same distance and gas price, the cost would be $900, illustrating significant potential savings.

Understanding fuel costs aids consumers by revealing which car could be more economical in the long run.

Break-even Analysis

Break-even analysis is a crucial tool in the decision-making process, especially for large purchases like cars. By analyzing the time it will take for a less expensive or more efficient choice to compensate for initial cost differences, consumers can make data-driven decisions.
To conduct a break-even analysis, one needs to calculate how long it will take for savings to outweigh initial differences in price. 1. Determine the initial cost difference between two choices. For example, Car B costs \(4,000 more than Car A.2. Compute the annual savings by choosing the more efficient option. For instance, if Car B saves \)400 yearly compared to Car A.3. Divide the initial cost difference by the annual savings to find the break-even point.In mathematical terms: \[ \text{Time to Break Even} = \frac{\text{Initial Cost Difference}}{\text{Annual Savings}} \]Using our example: \[ \frac{4000}{400} = 10 \]Thus, it takes 10 years for Car B's efficiency to surpass the initial cost difference.

Car Insurance Costs

Car insurance is a major recurring expense for car owners and significantly affects overall cost calculations in consumer decision-making.
Insurance rates depend on factors like the car model, the owner's driving history, and location. Consider two options and their insurance aspects:

• **Car A's Insurance**: Costs $1,000 annually.
• **Car B's Insurance**: Costs $1,200 annually, slightly higher due to the vehicle's value or risk profile.

Understanding these insurance costs helps provide a clear picture of each car's total annual expenses. It is important to compare these over the lifespan of the vehicle to assess overall affordability and long-term financial commitments.
Proper insurance analysis ensures that consumers are aware of varying insurance premiums related to different vehicles, contributing to a holistic view of the total cost of car ownership.