Problem 82 - Precalculus

Question

Use the following tables. The following table gives fuel and electric requirements per mile associated with gasoline and electric automobiles: $$\begin{array}{|l|c|c|}\hline & \begin{array}{c} \text { Numeer of } \\\\\text { GatLons/Mile }\end{array} & \begin{array}{c}\text { Numeer of } \\\\\text { kW-hr/Mile }\end{array} \\\\\hline \text { SUV full size } & 0.06 & 0 \\ \hline \text { Hybrid car } & 0.02 & 0.1 \\\\\hline \text { Electric car } & 0 & 0.3 \\\\\hline\end{array}$$ The following table gives an average cost for gasoline and electricity: $$\begin{array}{|l|c|}\hline \text { Cost per gallon of gasoline } & \$ 3.80 \\\\\hline \text { Cost per kW-hr of electricity } & \$ 0.05 \\\\\hline\end{array}$$ Environment. Assume you drive 12,000 miles per year. What are the yearly costs associated with driving the three types of cars in Exercise \(81 ?\)

Step-by-Step Solution

Answer

Yearly cost: SUV = $2,736, Hybrid car = $972, Electric car = $180.

Step 1: Calculate Gasoline Cost for SUV

For the SUV, we find the yearly gasoline cost by multiplying the number of gallons per mile (0.06 gallons/mile) by the total miles driven per year (12,000 miles), and then by the cost per gallon of gasoline (\(3.80). \[ \text{Cost}_{SUV} = 0.06 \times 12,000 \times 3.80 = 2,736 \]So, the yearly cost for the SUV is \)2,736.

Step 2: Calculate Total Cost for Hybrid Car

For the hybrid car, we calculate both the gasoline cost and the electricity cost. - **Gasoline Cost:** Mulitply the number of gallons per mile (0.02) by the total miles driven (12,000) and the cost per gallon of gasoline (\(3.80). \[ \text{Gas Cost}_{Hybrid} = 0.02 \times 12,000 \times 3.80 = 912 \]- **Electricity Cost:** Multiply the number of kW-hr per mile (0.1) by the total miles (12,000) and the cost per kW-hr (\)0.05). \[ \text{Electricity Cost}_{Hybrid} = 0.1 \times 12,000 \times 0.05 = 60 \]- **Total Cost for Hybrid Car:** Add the gasoline and electricity cost. \[ \text{Total Cost}_{Hybrid} = 912 + 60 = 972 \]So, the yearly cost for the hybrid car is $972.

Step 3: Calculate Electricity Cost for Electric Car

For the electric car, we only need to calculate the electricity cost. Multiply the number of kW-hr per mile (0.3) by the total miles driven (12,000) and the cost per kW-hr (\(0.05). \[ \text{Cost}_{Electric} = 0.3 \times 12,000 \times 0.05 = 180 \]So, the yearly cost for the electric car is \)180.

Key Concepts

Fuel Efficiency

Fuel efficiency is crucial when analyzing the cost of driving different types of vehicles over a year. It refers to how effectively a vehicle uses energy to travel a particular distance. Fuel efficiency can significantly impact the overall cost because it determines how much fuel, whether gasoline or electricity, a car requires.
For vehicles using gasoline, fuel efficiency is usually measured in gallons per mile. The lower the number, the less fuel is needed, leading to cost savings. For example, in the exercise, an SUV that consumes 0.06 gallons per mile will incur higher costs compared to a hybrid car using only 0.02 gallons per mile.

• A well-maintained and efficient engine can improve fuel efficiency.
• Regularly inflating tires to the proper pressure can also help.

Electric vehicles measure fuel efficiency in kilowatt-hours (kW-hr) per mile, emphasizing the amount of electrical energy required to travel a mile. As with gasoline, a lower number indicates better efficiency.
Therefore, owning a car with high fuel efficiency will reduce the total cost spent on energy throughout the year.

Hybrid Vehicles

Hybrid vehicles combine gasoline engines with electric motors to improve fuel efficiency and minimize emissions. They are designed to switch seamlessly between gasoline and electric power or use both in tandem.
For our exercise, the ability of hybrid vehicles to use electricity in addition to gasoline means analyzing the cost involves factors for both fuel types. The hybrid car in the exercise uses 0.02 gallons per mile and 0.1 kW-hr per mile, which leads to distinct costs associated with both gasoline and electricity.

• Gasoline costs are calculated similarly to traditional vehicles based purely on gas consumption.
• Electric costs require calculating kilowatt-hours used, multiplied by the cost per kW-hr, which can be economical depending on electricity rates.

The advantage of hybrid cars is their flexibility in reducing costs by making use of cheaper electricity, especially for short commutes or city driving. Many hybrid car owners enjoy the cost savings from utilizing electric power, especially when gas prices soar.

Electric Vehicles

Electric vehicles (EVs) rely solely on electric power, having no gasoline engine. They produce zero emissions during operation and can be a cost-effective option with low energy prices. In the original exercise, cost analysis of an electric vehicle showed it required 0.3 kilowatt-hours per mile.
The appeal of electric cars lies in their efficiency and lower operational costs compared to gas-powered cars. The total yearly cost for electricity, given it's relatively inexpensive, can be much lower than fueling traditional vehicles

• Home charging stations can lead to further savings, especially if paired with solar energy systems.
• Many regions offer incentives for EV ownership, including reduced registration fees and tax credits.

Although the initial purchase price of electric vehicles can be higher, the long-term operational savings often balance out the investment. With advancing technology, EVs are becoming more accessible options for many drivers looking to reduce their environmental impact and save on yearly costs.