Question

How long will it take the U.S. annual oil consumption to double if it is increasing exponentially at a rate of \(7.0 \%\) per year?

Short answer

It will take approximately 9.9 years for the U.S. annual oil consumption to double at a 7.0% annual increase rate.

Step-by-step solution

Understanding the Problem

We are dealing with exponential growth where the quantity at any time can be described by the equation \( P(t) = P_0 e^{rt} \). In this equation, \( P(t) \) is the amount at time \( t \), \( P_0 \) is the initial amount, \( r \) is the growth rate, and \( t \) is the time in years. Here, we need to find the value of \( t \) when the U.S. oil consumption doubles, which means \( P(t) = 2P_0 \).

Setting Up the Equation

Using the exponential growth formula, we can set up the equation to find the time \( t \) it takes to double. Since \( P(t) = 2P_0 \), we have \( 2P_0 = P_0 e^{0.07t} \).

Solving the Equation

To solve for \( t \), we first divide both sides of the equation by \( P_0 \), which gives us \( 2 = e^{0.07t} \). Now, we take the natural logarithm of both sides to get \( \ln(2) = 0.07t \). Finally, we solve for \( t \) by dividing both sides by 0.07, which gives us \( t = \frac{\ln(2)}{0.07} \).

Calculating the Time

Using a calculator to find the natural logarithm of 2, we get \( \ln(2) \approx 0.6931 \). Dividing this by 0.07, the time \( t \) is approximately \( t \approx \frac{0.6931}{0.07} \approx 9.9 \) years.

Key concepts

Exponential Growth Formula

The concept of exponential growth is pivotal in understanding how quantities increase over time at a rate that is proportional to their current value. It describes a situation where the growth speed of a mathematical function is directly proportional to the value of the function at any given time. For instance, in scenarios such as population growth, nuclear chain reactions, or, as in our exercise, the U.S. annual oil consumption, exponential growth is how we model expansion that rapidly increases over time.

The exponential growth formula is ubiquitously represented as:
\[ P(t) = P_0 e^{rt} \]
Here, \( P(t) \) represents the quantity at time \( t \), \( P_0 \) is the initial quantity, \( r \) is the growth rate, and \( e \) is the base of the natural logarithm, an important mathematical constant approximately equal to 2.71828. When we speak of 'doubling time,' we are typically trying to find the value of \( t \) that causes \( P(t) \) to be twice the value of \( P_0 \), a common query in fields such as finance, biology, and environmental science.

Application of this formula to real-world situations helps us make predictions and plan for the future based on current trends. The clear expectation in our exercise is determining when the consumption of a resource, like oil, will become twice as high as it is now, given that it increases by a fixed percentage each year.

Natural Logarithm

The natural logarithm, denoted as \( \text{ln} \), is the logarithm to the base of the mathematical constant \( e \), where \( e \) is an irrational and transcendental number, often approximated as 2.71828. In essence, the natural logarithm of a number \( x \) is the exponent to which \( e \) must be raised to produce that number \( x \). Thus, given an equation \( e^y = x \), then \( y = \text{ln}(x) \).

Natural logarithms are particularly useful in solving equations involving exponential growth, where they allow us to isolate the variable of interest, for example, time in the doubling equation. This utility is evident in solving the textbook exercise, where we took the natural logarithm of both sides to transform the exponential equation into a linear one, enabling us to solve for \( t \).

The relationship and transition between exponential functions and logarithms are fundamental in various scientific and mathematical fields, including compound interest calculations in finance, radioactive decay in physics, and the analysis of growth patterns in biology.

Rate of Increase

The rate of increase, often expressed as a percentage, signifies the speed at which a quantity grows over a specified period. In the case of our textbook problem, the U.S. oil consumption is said to be increasing at a rate of 7.0% per year. This rate becomes the \( r \) in our exponential growth formula and is a crucial factor in determining how quickly a quantity will double, triple, or reach any predefined level.

In mathematical terms, a rate of 7.0% per year translates to a growth factor of 0.07 when used in our exponential equation. It's important to understand that even seemingly small percentage rates can lead to significant increases over time due to the nature of exponential growth. This concept allows us to not only predict future values, but to also calculate the period it will take for a certain growth-driven event to occur, as demonstrated in the solution that calculates the doubling time of the aforementioned oil consumption.

Recognizing the implications of the rate of increase can be critical in areas such as resource management, financial planning, and understanding population dynamics. It underscores the impact of consistent growth percentages over time and why such rates are critical to watch and understand whether they pertain to interest on investments or the usage of natural resources.