Question

The world's total petroleum reserve is estimated at \(2.0 \times 10^{22}\) joules [a joule (J) is the unit of energy where \(\left.1 \mathrm{~J}=1 \mathrm{~kg} \cdot \mathrm{m}^{2} / \mathrm{s}^{2}\right]\). At the present rate of consumption, \(1.8 \times 10^{20}\) joules per year (J/yr), how long would it take to exhaust the supply?

Short answer

It will take approximately 111 years to exhaust the supply.

Step-by-step solution

Write Down the Given Quantities

We are given that the total petroleum reserve is \(2.0 \times 10^{22}\) joules. The consumption rate is \(1.8 \times 10^{20}\) joules per year.

Set Up the Division Problem

To find out how many years it will take to exhaust the petroleum reserve, we need to divide the total reserve by the annual consumption rate. The formula is: \[ \text{Time to exhaust} = \frac{\text{Total Reserve}}{\text{Consumption Rate}} \] Substitute the values into the formula.

Perform the Calculation

Substitute the given values into the formula:\[ \text{Time to exhaust} = \frac{2.0 \times 10^{22}}{1.8 \times 10^{20}} \]To simplify: First, divide the coefficients: \( \frac{2.0}{1.8} \approx 1.11 \)Next, subtract the exponents when dividing exponential terms: \(22 - 20 = 2\) Thus,\[ \text{Time to exhaust} = 1.11 \times 10^{2} \] Therefore, it will take approximately 111 years.

Key concepts

Petroleum Reserve

When we talk about a petroleum reserve, we're referring to the total amount of stored energy that is available in the form of petroleum. This reserve is a crucial component for understanding energy resources on a global scale. The unit of measurement used here is joules, specifically highlighting that the reserve is estimated at a massive \(2.0 \times 10^{22}\) joules. By appreciating the sheer size of this number, we grasp the enormity of our petroleum resources.

Petroleum reserves are significant due to their role in powering industries, transportation, and daily life conveniences. They are considered non-renewable resources, meaning they do not replenish at a rate that meets consumption needs. Hence, understanding these reserves allows us to plan for sustainable energy use and transition.

Joules

The unit 'joules' represents energy and is fundamental for discussing energy reserves and consumption. One joule (\(1 \, J\), specifically, is defined as the amount of energy transferred when applying a force of one newton over a one-meter distance. Mathematically, it is expressed as \(1 \, J \, = \, 1 \, kg \, \cdot \, m^2/s^2\), showcasing how energy involves both force and movement.

Joules are ubiquitously applied in various energy discussions, such as electricity, heat, and mechanical work. By using joules, we create a standardized way to measure and communicate energy quantities, whether small (like a light bulb’s consumption) or vast (like petroleum reserves). To ensure meaningful comparisons and calculations, it's important to convert other energy units to joules, maintaining consistency in energy discussions.

Exponential Division

Exponential division comes into play when handling large numbers, such as those in scientific notation. This approach simplifies calculations and allows us to easily manage huge figures like the power of ten. In this scenario, both the petroleum reserve and the consumption rate are represented using scientific notation: \(2.0 \times 10^{22}\) and \(1.8 \times 10^{20}\), respectively.

To divide numbers in scientific notation, you first divide the coefficients (2.0 by 1.8) and then subtract the exponents (22 minus 20). This simplifies to multiplying the divided coefficient by 10 raised to the power of the resulting exponent, giving an easy-to-interpret number. This method keeps things simpler without having to deal with impractically large numbers directly.

Resource Depletion Calculation

Resource depletion calculation is crucial for understanding how long a resource will last at the current consumption level. It illuminates our need to manage resources more sustainably. In this problem, knowing that the petroleum total reserve is \(2.0 \times 10^{22}\) joules and that the consumption rate is \(1.8 \times 10^{20}\) joules per year allows us to calculate how many years until this resource is exhausted.

By setting up a division problem, we can determine the lifespan of the petroleum reserve by dividing the total reserve by the annual consumption rate. The result, \(1.11 \times 10^{2}\), translates to approximately 111 years. This calculation offers a tangible timeline for resource usage, underscoring the importance of efficient and conscious consumption to extend the life of our finite resources.