Question

A gallon of gasoline contains about \(37 \mathrm{kWh}\) of energy and costs about \(\$ 4\), while a typical AA battery holds about \(0.003 \mathrm{kWh}\) and costs about \(\$ 0.50\) each, in bulk. By what factor are batteries more expensive, for the same amount of energy?

Short answer

Short Answer: AA batteries are approximately 3083 times more expensive than gasoline for the same amount of energy.

Step-by-step solution

Calculate cost per kWh of gasoline

To find the cost per kWh for gasoline, we need to divide the cost of one gallon of gasoline by the energy it contains: Cost per kWh (Gasoline) = \(\frac{\text{Cost of one gallon}}{\text{Energy in one gallon}}\) Cost per kWh (Gasoline) = \(\frac{\$4}{37 \mathrm{kWh}}\)

Calculate cost per kWh of AA battery

To find the cost per kWh for an AA battery, we need to divide the cost of one AA battery by the energy it contains: Cost per kWh (AA Battery) = \(\frac{\text{Cost of one AA battery}}{\text{Energy in one AA battery}}\) Cost per kWh (AA Battery) = \(\frac{\$0.50}{0.003 \mathrm{kWh}}\)

: Step 3: Calculate the factor by which batteries are more expensive

: To find the factor by which batteries are more expensive, we need to divide the cost per kWh of an AA battery by the cost per kWh of gasoline: Factor = \(\frac{\text{Cost per kWh (AA Battery)}}{\text{Cost per kWh (Gasoline)}}\) Factor = \(\frac{\frac{\$0.50}{0.003 \mathrm{kWh}}}{\frac{\$4}{37 \mathrm{kWh}}}\) Factor = \(\frac{(\$0.50)(37 \mathrm{kWh})}{(0.003 \mathrm{kWh})(\$4)}\) Now, let's perform the calculations: Factor = \(\frac{(0.50)(37)}{(0.003)(4)}\) Factor ≈ 3083.33 So batteries are approximately 3083 times more expensive than gasoline for the same amount of energy.

Key concepts

kWh Calculation

The kilowatt-hour (kWh) is a standard unit of energy used to measure how much electricity you are using. It is crucial to understand how to calculate kWh when comparing different energy sources.

To find the kWh, you need to know two things: the power consumption in kilowatts (kW) of the device or source, and the amount of time it is used in hours. The formula is simple: multiply the kW by the number of hours, which gives the energy consumption in kWh.

In the exercise, we are given certain energy values directly. For gasoline, it's stated as 37 kWh per gallon, and for AA batteries, it's 0.003 kWh per battery. This allows us to directly calculate and compare costs without needing further conversion. This step is fundamental in determining how efficient and costly different sources of energy are.

Cost Analysis

Cost analysis helps us understand the financial implications of using various energy sources. It is all about determining how much it costs to derive a certain amount of energy from different sources.

In the exercise, we learn to calculate the cost per kWh for each energy source. Here’s the breakdown:

• For gasoline: Divide the cost of a gallon by its energy content. This gives: Cost per kWh (Gasoline) = \( \frac{\\(4}{37 \text{kWh}} \).
• For an AA battery: Divide the cost of one battery by its energy content. This gives: Cost per kWh (AA Battery) = \( \frac{\\)0.50}{0.003 \text{kWh}} \).

These calculations allow us to compare how much each energy source costs for the same unit of energy (kWh). This is critical for making cost-effective decisions, especially when managing budgets in personal or industrial settings.

Energy Sources

Understanding different energy sources and their characteristics help us make informed choices about which are most suitable for our needs. Energy sources can vary widely in cost, availability, and how they deliver power.

In this example, gasoline and AA batteries are compared in terms of cost efficiency. Gasoline is often used for vehicles and engines, having a relatively high energy content per unit volume, making it a common choice for high energy demand.

AA batteries, on the other hand, are convenient for small electronic devices due to their portability but come at a much higher cost per kWh as shown in the exercise. This highlights the trade-off between cost and convenience. Knowing these differences helps individuals and businesses select the right energy sources for their applications, balancing both cost and efficiency.