Question

Use either the formulas or the universal growth and decay curves, as directed by your instructor.The oil consumption of a certain country was 12.0 million barrels in 2010 , and it grows at an annual rate of \(8.30 \% .\) Find the oil consumption by the year \(2015 .\)

Short answer

The oil consumption by the year 2015 is approximately 17.8 million barrels.

Step-by-step solution

Identifying the Initial Quantity

Determine the initial quantity of oil consumption. In this problem, the initial quantity, often represented as (umber{0}), is the consumption level at the starting year, which is 12.0 million barrels in 2010.

Determining the Growth Rate

Identify the annual growth rate of oil consumption. The growth rate is given as 8.30%, which is 0.083 in decimal form.

Calculating the Number of Periods

Calculate the number of periods (years) over which the oil consumption grows. Since we want the consumption for 2015 and we start from 2010, the number of periods is 2015 minus 2010, which is 5 years.

Using the Exponential Growth Formula

Apply the exponential growth formula to calculate the future amount of oil consumption: (umber{(t)}) = (umber{0})e^{rt}. Here, (umber{(t)}) is the amount after t periods, (umber{0}) is the initial amount, r is the growth rate in decimal form, and t is the time in years.

Plugging in the Values

Substitute the known values into the formula: (umber{(5)}) = 12.0 * e^{0.083 * 5}.

Calculating the Exponential Function

Calculate the value of e raised to the power of (0.083 * 5) to find the growth factor.

Calculating the Oil Consumption for 2015

Multiply the initial oil consumption by the growth factor to find the oil consumption in 2015.

Key concepts

Initial Quantity

Understanding the initial quantity in exponential growth problems is critical. The initial quantity, also known as the initial value or amount, is the starting level from which growth or decay is measured. In the case of the oil consumption problem, this refers to the amount of oil that the country consumed at the beginning of the observation period, which is 12.0 million barrels in the year 2010. This figure serves as the baseline for projecting future consumption using the exponential growth model.

Grasping the importance of the initial quantity leads to a better understanding of how exponential growth will impact the subject in question over time. It lays the foundation for the subsequent steps that, when followed correctly, yield accurate projections or predictions.

Growth Rate

The growth rate is a crucial variable that determines the speed at which the quantity is growing. Expressed usually as a percentage, it needs to be converted into decimal form to be used in mathematical formulas. In our example, the annual growth rate of oil consumption is 8.30%, which we convert to 0.083 in decimal. The growth rate directly influences how quickly the oil consumption will increase year over year.

It's important to note that growth rates can vary depending on the context and what is being measured. As such, accurately determining and interpreting growth rates is an essential skill in a variety of disciplines including finance, population study, and resource management.

Exponential Growth Formula

The exponential growth formula is a mathematical representation used to calculate the amount of growth over time. It is often written as \( Q(t) = Q_0e^{rt} \), where \( Q(t) \) is the quantity after time t, \( Q_0 \) is the initial quantity, \( r \) is the growth rate in decimal form, and \( e \) is the base of the natural logarithm, approximately equal to 2.71828. This formula provides a precise method for forecasting exponential growth, as it accounts for the compound nature of growth as time progresses.

For students and professionals alike, mastering this formula is key to solving problems involving exponential growth, such as population studies, financial investments, and, in our case, projecting oil consumption.

Calculating Exponential Growth

Calculating exponential growth involves applying the exponential growth formula to find the expected future quantity of the subject in question. By following a step-by-step approach, you begin with identifying the initial quantity and growth rate, which are then plugged into the formula. The process also includes calculating the number of periods over which the growth occurs — in our example, the 5 years between 2010 and 2015. Once these values are collected and substituted into the formula, you compute the exponential function and multiply the initial quantity by the growth factor to reach the final projection.

Understanding the mechanics of this calculation helps individuals to predict trends and make informed decisions based on the projected outcomes. Students will often use calculators or software to compute the value of \( e \) raised to the power of the growth rate multiplied by time, as this step requires precise mathematical computation.

Oil Consumption Projection

Using exponential growth to project oil consumption can inform government policies and economic planning. Considering the initial oil consumption and applying the growth rate over a specified period, stakeholders can estimate future needs and plan accordingly. For instance, the projection formula used in our exercise predicts the consumption of oil in 2015 based on the given data from 2010. Such projections help in understanding the trajectory of resource usage and highlight the potential need for sustainable measures or alternative energy sources.

It's significant to acknowledge that these projections, while useful, are based on the assumption that growth rates remain constant over time. In reality, numerous factors could alter consumption patterns, including technological advancements, changes in regulation, or geopolitical events, thus it's always beneficial to approach such projections with an understanding of their inherent limitations.