Question

If a resource is being used up at a rate that increases exponentially, the time it takes to exhaust the resource (called the exponential expiration time, EET) is $$\mathrm{EET}=\frac{1}{n} \ln \left(\frac{n R}{r}+1\right)$$ where \(n\) is the rate of increase in consumption, \(R\) the total amount of the resource, and \(r\) the initial rate of consumption. If we assume that the United States has oil reserves of \(207 \times 10^{9}\) barrels and that our present rate of consumption is \(6.00 \times 10^{9}\) barrels/yr, how long will it take to exhaust these reserves if our consumption increases by \(7.00 \%\) per year?

Short answer

Using the given formula and substituting the values for the rate of increase in consumption, total resource amount, and initial consumption rate, it will take approximately 17.57 years to exhaust the oil reserves if consumption increases by 7% per year.

Step-by-step solution

Understand and Identify Given Variables

Identify the given variables: the rate of increase in consumption (n), the total amount of the resource (R), and the initial rate of consumption (r). In this case, we have n as 7%, which is 0.07 when expressed as a decimal, R as 207 x 10^9 barrels, and r as 6.00 x 10^9 barrels/year.

Substitute the Known Values into the EET Formula

Substitute these values into the Exponential Expiration Time (EET) formula: \(\mathrm{EET}=\frac{1}{n} \ln \left(\frac{n R}{r}+1\right)\).

Calculate the Exponential Expiration Time (EET)

Perform the calculation with the substituted values:\(\mathrm{EET}=\frac{1}{0.07} \ln \left(\frac{0.07 \cdot 207 \times 10^{9}}{6.00 \times 10^{9}}+1\right)\).

Resolve the Numerator in the Logarithm

First, resolve the numerator inside the logarithm to simplify it: \(\frac{0.07 \cdot 207 \times 10^{9}}{6.00 \times 10^{9}} = 2.42\). Therefore, the expression inside the logarithm becomes \(2.42+1\).

Calculate the Logarithmic Function

Find the natural logarithm of the sum inside the logarithm: \(\ln(2.42+1)\) or \(\ln(3.42)\).

Complete the Calculation

Calculate the full expression to find the EET: \(\mathrm{EET}= \frac{1}{0.07} \ln(3.42)\). The natural logarithm of 3.42 is approximately 1.23.

Final Calculation

Multiply the result of the natural logarithm by the reciprocal of the rate of increase, which gives us the final EET: \(\mathrm{EET}= \frac{1.23}{0.07}\), which is approximately 17.57 years.

Key concepts

Exponential Decay

The concept of exponential decay is central to understanding phenomena where quantities decrease at a rate proportional to their current value. This type of decay is ubiquitous in various fields such as radioactive decay in physics, depreciation in economics, and, as in our example, the consumption of resources.

Mathematically, exponential decay can be described by the formula
\[ A(t) = A_0 e^{-kt} \]
where \( A(t) \) is the amount of substance remaining after time \( t \), \( A_0 \) is the initial amount, \( e \) is the base of the natural logarithm, and \( k \) is the decay constant. The formula reflects how the quantity diminishes over time at a rate that is 'exponentially' proportional to its current amount. Speaking of the rate of consumption for a resource, the decay constant would determine how quickly the resource depletes, which is critical to sustainability and planning.

Natural Logarithm

The natural logarithm is an essential mathematical function that is often represented as \( \ln(x) \). Its inverse is the exponential function \( e^x \), where \( e \) is approximately equal to 2.71828. In the context of exponential decay and, more specifically, the Exponential Expiration Time (EET) equation, the natural logarithm is used to 'unpack' the exponential function.

Understanding how to work with the natural logarithm is crucial when analyzing exponential growth or decay. For example, it can help determine the time it takes for a resource to be exhausted, as shown in our initial problem. The property that \( \ln(e^x) = x \) makes it particularly useful for solving equations where the variable of interest is an exponent. Additionally, the natural logarithm has a continuous growth rate, making it inherently linked with the concept of rate of consumption.

Rate of Consumption

Rate of consumption refers to the speed at which a resource is used. In our exercise, the rate increases exponentially, implying that as time progresses, the consumption speed accelerates at a rate that is proportional to its current level.

This exponential increase can be due to various factors, such as technological advancements leading to greater resource usage or a population growth resulting in higher demand. Understanding the rate of consumption is pivotal for predicting when a resource might run out and for planning sustainable usage. In the EET formula, the initial rate of consumption \( (r) \) and the rate of increase in consumption \( (n) \) are critical to estimating the time until the resource is exhausted. The formula incorporates the natural logarithm, which helps transform the exponential rate into a linear timeframe, making it easier to understand and calculate.

Mathematical Modeling

Mathematical modeling involves creating equations to represent real-world situations. It's an invaluable tool for making predictions about future events based on current knowledge. In our case, the EET formula is a mathematical model that helps predict the exhaustion time of a resource under exponential consumption growth.

Modeling requires not only the appropriate formula but also accurate values for variables, such as the initial rate of consumption, the rate of increase, and total resources. Students using these models need a solid understanding of the underlying principles, like exponential decay and logarithms, to accurately interpret results. Such models are simplifications of reality, so it is important to recognize their limitations and potential for refinement as new data becomes available. Nevertheless, they are central to strategic planning in resource management, environmental science, economics, and many other disciplines.