Question

In some cities, taxicabs run on liquefied propane fuel instead of gasoline. This practice extends the lifetime of the vehicle and produces less pollution. Given that it costs about 2000 to modify the engine of a taxicab to run on propane and that the cost of gasoline and liquid propane are 3.50 per gallon and 2.50 per gallon, respectively, make reasonable assumptions and figure out how many miles a taxi would have to go so that the decreased fuel cost would balance the added cost of modifying the taxi's motor.

Short answer

The taxi must drive \(\frac{2000}{f}\) miles, where "f" is the fuel consumption in gallons per mile.

Step-by-step solution

Understanding the Problem

We need to find out how many miles a taxi must drive so that the savings on fuel costs from using propane instead of gasoline balance out the $2000 modification cost.

Define Variables

Let the vehicle consume fuel at a rate of "f" gallons per mile. Let "m" be the number of miles driven. The cost to modify the taxi is $2000.

Set Up the Equations

The cost to run the taxi on gasoline over "m" miles is \(3.50 \times f \times m\). The cost to run the taxi on propane over "m" miles is \(2.50 \times f \times m + 2000\).

Equate the Costs

For the costs to balance, \(3.50 \times f \times m = 2.50 \times f \times m + 2000\).

Solve the Equation

Subtract \(2.50 \times f \times m\) from both sides: \(1.00 \times f \times m = 2000\). Simplify to find \(m = \frac{2000}{f}\).

Interpret the Result

The number of miles, "m," required depends inversely on the fuel consumption "f," meaning the more fuel efficient the vehicle is (smaller "f"), the more miles it needs to travel.

Key concepts

Propane Fuel

Propane is a type of liquefied petroleum gas (LPG) that is commonly used as a fuel for heating, cooking, and in some cases, powering vehicles like taxicabs. What makes propane an attractive alternative to gasoline? Well, there are several reasons:
- **Cost Efficiency**: Propane generally costs less per gallon compared to gasoline, making it a more economical choice for fueling vehicles over long distances. This price difference can accumulate into significant savings over time.
- **Availability**: Propane is widely available, which makes it convenient for vehicle owners to access fuel easily.
- **Storage**: As a liquid, propane is compressed, allowing for more energy to be stored in a smaller volume than traditional gasoline.
Understanding how propane operates as a fuel source is essential as it underpins our ability to evaluate its practicality for modifying vehicles like taxicabs and realizing potential savings.

Pollution Reduction

One of the main advantages of using propane over gasoline is that it results in less air pollution. Vehicles powered by propane can emit fewer harmful substances into the atmosphere compared to those running on gasoline. Key environmental benefits include:
- **Reduction in Greenhouse Gas Emissions**: Propane combustion produces fewer greenhouse gases, contributing less to global warming than gasoline.
- **Lower Emissions of Harmful Pollutants**: Compared to gasoline, propane emits lower levels of carbon monoxide, nitrogen oxides, and particulate matter. This is beneficial for urban areas with high vehicle density, where air quality can be a significant health concern.
Therefore, switching to propane not only aids in reducing individual costs but also supports efforts for cleaner air and a healthier environment. It's a win-win for both economic and ecological reasons.

Fuel Cost Analysis

Fuel cost analysis is a critical aspect when considering a switch to propane for powering vehicles. To understand its savings, consider these factors:
- **Price Per Gallon**: Propane is cheaper by approximately one dollar per gallon compared to gasoline. With gasoline priced at $3.50 per gallon and propane at $2.50 per gallon, the disparity can lead to substantial savings, especially for vehicles driven extensively.
- **Modification Costs**: There is an upfront cost of about $2000 to modify a vehicle engine to run on propane. Although this is a significant expense, the ongoing savings in fuel costs can offset it over time.
To break even on the modification cost, it's crucial to calculate the mileage at which the propane fuel savings match the initial investment. This involves analyzing the fuel consumption rate and how efficiently the vehicle uses fuel to achieve the necessary balance.

Engine Modification

Modifying a taxicab engine to run on propane involves specific mechanical changes. This transformation is necessary for the vehicle to utilize propane efficiently. Important considerations include:
- **Conversion Costs**: Typically, an investment of around $2000 is required for engine modification. This cost includes changes to the engine and potential changes to the fuel storage system to safely accommodate propane.
- **Efficiency Considerations**: The vehicle's fuel efficiency, dictating how many gallons per mile it consumes, plays a crucial role in assessing how quickly the conversion cost can be recuperated through fuel savings.
- **Regulatory and Safety Compliance**: Modifications must adhere to safety regulations and standards to ensure that the vehicle operates safely under new fuel conditions.
Understanding these aspects of engine modification helps in preparing for a switch to propane, balancing initial costs with long-term benefits.