Problem 29 - Technical Mathematics with Calculus

Question

Energy Consumption: If the U.S. energy consumption is \(7.00 \%\) higher each year, by what factor will the energy consumption have increased after 10.0 years?

Step-by-Step Solution

Answer

After 10 years, the energy consumption will have increased by a factor of approximately 1.07^10, which equals about 1.967.

Step 1: Understanding the Problem

We need to determine the factor by which the U.S. energy consumption increases after 10 years given that it rises by 7.00% each year. This is an application of exponential growth.

Step 2: Calculate the Annual Growth Factor

To find the annual growth factor, we convert the percentage increase to decimal form and add it to 1. This is because the energy consumption is growing, not just changing. Growth factor = 1 + percentage increase (in decimal form) = 1 + 0.07 = 1.07.

Step 3: Applying the Growth Factor Over 10 Years

To find out by what factor the energy consumption will increase over 10 years, raise the annual growth factor to the power of 10. Factor after 10 years = (Annual growth factor)^(Number of years) = 1.07^10.

Step 4: Calculate the Result

Using a calculator, compute the value of 1.07 raised to the power of 10. This will give us the factor by which energy consumption has increased after 10 years.

Key Concepts

Energy Consumption

When discussing energy consumption, we're referring to the amount of energy used by a nation, organization, or device. In our everyday lives, this could range from the electricity needed to power our homes to the fuel used to run vehicles. Energy consumption is measured over a period, often annually, and is a vital indicator of economic activity, efficiency, and technological advancement.

Understanding how energy consumption grows over time is crucial, particularly in light of our increasing global population and industrial development. A rise in energy consumption can signify economic growth, but it also raises questions on sustainability and environmental impact. To tackle these challenges, analyzing energy usage trends and foreseeing future demands becomes essential, often employing mathematical models such as exponential growth which are central to our exercise here.

Percentage Increase

The percentage increase is a common way to express the change in a quantity over time. It is calculated as the difference between the new and original quantities, divided by the original quantity, and then multiplied by 100 to convert to a percentage.

For example, when we say the U.S. energy consumption increases by 7.00% each year, it means that for each year, the new amount of energy consumed is 107% of the previous year's amount. Mathematically, the percentage increase helps us understand the rate at which something is growing; it is a straightforward way to communicate changes, especially when the actual amounts involved are large or complex, as with national energy consumption rates.

Annual Growth Factor

The annual growth factor reflects how much a quantity grows by each year, and it is a key concept in understanding exponential growth. It is the multiplier used to find a future value of a quantity experiencing steady percentage increase over time. To calculate it, you translate the percentage increase into a decimal and add it to 1; the sum is your growth factor. For example, with a 7.00% increase, the growth factor is 1.07.

To project future growth, you raise this growth factor to the power corresponding to the number of years you’re looking at. So, in our exercise, to determine the factor by which U.S. energy consumption will have increased after 10 years, you calculate as follows: \(1.07^{10}\). This formula provides a simple yet powerful way to predict exponential growth over time.