Question

You want to decide whether you should drive your car to work or take the train. You know the one-way distance from your home to your place of work, and the fuel efficiency of your car (in miles per gallon). You also know the one- way price of a train ticket. You assume the cost of gas at \(\$ 4\) per gallon, and car maintenance at 5 cents per mile. Write an algorithm to decide which commute is cheaper.

Short answer

Calculate and compare car travel cost and train cost to decide which is cheaper.

Step-by-step solution

Define the known values

Let's start by identifying the known values from the problem. Let the one-way distance be \(d\) miles and the fuel efficiency of the car be \(e\) miles per gallon. The one-way train ticket cost is \(c_t\) dollars. Gas cost is \(\\(4\) per gallon, and car maintenance is \(\\)0.05\) per mile.

Calculate the car travel costs

To calculate the cost of driving the car one way, we need to find the gas cost and the maintenance cost. - **Gas Cost**: \( \text{Gas Cost} = \left( \frac{d}{e} \right) \times 4 \)- **Maintenance Cost**: \( \text{Maintenance Cost} = d \times 0.05 \)**Total Car Cost** = Gas Cost + Maintenance Cost = \( \left( \frac{d}{e} \right) \times 4 + d \times 0.05 \).

Compare costs

Now that we have the formula for the car cost and know the cost for the train, compare the costs to determine which commute is cheaper:- If **Total Car Cost** < \(c_t\), then driving the car is cheaper.- If **Total Car Cost** > \(c_t\), then taking the train is cheaper.- If **Total Car Cost** = \(c_t\), both options cost the same.

Key concepts

Cost Analysis

In making any transportation decision, the cost analysis becomes crucial. The goal is to break down the potential expenses associated with each commuting method. For this problem, we're looking at the costs associated with driving a car versus taking a train.

### Understanding the Cost Components for the CarWhen analyzing the costs of driving, two main components need to be considered: the gas cost and the maintenance cost.

• **Gas Cost**: This can be calculated using the formula \( \left( \frac{d}{e} \right) \times 4 \), where \(d\) is the distance, \(e\) is the fuel efficiency (miles per gallon), and \(4\) is the cost per gallon.
• **Maintenance Cost**: This cost is added at a standard rate of \(0.05\) per mile, meaning you multiply the distance traveled by this rate to find the total maintenance expense.

Adding these components gives the total travel cost by car. A thorough cost analysis requires comparing this total with the known cost of the train ticket to assess the least expensive option.

Transportation Decision Making

After conducting a cost analysis, we proceed to the decision-making phase. This involves directly comparing the total costs of driving versus taking the train.

In this exercise, you need to determine the cheaper option for your commute.

• If the **Total Car Cost** (calculated from adding Gas Cost and Maintenance Cost) is less than the train ticket cost \(c_t\), then it's economically wiser to drive your car.
• If the **Total Car Cost** exceeds the price of the train ticket, then the train is the more affordable choice.
• In cases where they are equal, there is no cost advantage of one over the other; thus, the choice may rely on other factors such as convenience and time.

This kind of clear comparison is central to making an informed transportation decision, allowing you to optimize your commuting expenses efficiently.

Efficiency Calculations

Efficiency plays a key role in determining the best approach for a given task, especially when it involves resource usage. Here, we focus on two types of efficiencies: fuel efficiency and cost efficiency.

### Calculating Fuel EfficiencyFuel efficiency measures how far you can travel per unit of fuel. For this problem, it's defined in miles per gallon. It's used to calculate the gas cost by determining how many gallons are needed for the distance \(d\).### Calculating Cost EfficiencyCost efficiency in transportation decision-making means determining which option, car or train, gives you the best value for money. This involves comparing the costs calculated for each mode of transportation. It's about balancing the spending to achieve the most financially favorable outcome.When both efficiencies are considered, you get a complete picture of what is both resource and cost-efficient, helping you make smarter transportation decisions.