Question

Computer: Assuming that the present annual world oil consumption is \(17 \times 10^{9}\) barrels/yr, that this rate of consumption is increasing at a rate of \(5 \%\) per year, and that the total world oil reserves are \(1700 \times 10^{9}\) barrels, compute and print the following table:$$\begin{array}{ccc}\hline \text { Year } & \text { Annual Consumption } & \text { Oil Remaining } \\\\\hline 0 & 17 & 1700 \\\1 & 17.85 & 1682.15 \\\& & \\\\\cdot & & \\ & &\end{array}$$,Have the computation stop when the oil reserves are almost gone.

Short answer

Create a loop to calculate each year's consumption by increasing it by 5% from the previous year's consumption, and subtract that from the remaining reserves until the reserves are nearly depleted.

Step-by-step solution

Understand the Problem

We need to calculate the annual oil consumption and remaining oil reserves for each year. The annual consumption starts at 17 billion barrels and increases by 5% each year. The initial oil reserve is 1700 billion barrels. The calculation continues until the reserves are nearly depleted.

Set Up the Initial Values

Let's set the initial year as Year 0, the initial annual consumption as 17 billion barrels, and the initial oil reserve as 1700 billion barrels.

Establish the Recursive Relationships

For each subsequent year, the annual consumption will be 5% greater than the previous year's consumption. This can be calculated as: Annual consumption for next year = Annual consumption for current year * (1 + 5%). Simultaneously, the remaining oil reserve will decrease by the amount of the annual consumption. The remaining oil for the next year = Remaining oil for the current year - Annual consumption for the current year.

Create the Calculation Loop

Each year, increase the year count, calculate the new annual consumption based on the previous year's consumption increased by 5%, and calculate the remaining oil reserves by subtracting the annual consumption from the current reserves. This loop will continue until the remaining reserves are less than or equal to the annual consumption.

Compute the Table Values

Starting with year 0, use the formulas from Step 3 to calculate the annual consumption and remaining oil for each year. Append each year's data to the table and repeat until the oil is nearly gone.

Key concepts

Annual Oil Consumption Increase

Understanding the dynamics of annual oil consumption increase is crucial when discussing global energy resources. In our exercise, we begin with an annual consumption of 17 billion barrels, which is anticipated to grow by 5% each year. This percentage increment reflects factors such as economic development, population growth, and technological advancements that drive the demand for oil.

To put this into perspective, an annual consumption increase of 5% means that if we start with 17 billion barrels this year, the next year we will consume 17.85 billion barrels, and the following year we will consume even more. Over time, this increase compounds, which can be an eye-opener to the rate at which we are using this non-renewable resource. Students should be mindful to include this rate of increase year on year to accurately calculate future oil consumption.

Oil Reserves Depletion

The concept of oil reserves depletion touches on the finite nature of this vital energy resource. The exercise illustrates a starting point of 1700 billion barrels of oil in the world’s reserves. As we continue to consume oil at an increasing rate, these reserves are reduced annually.

The annual consumption subtracts from the remaining reserves, which leads to a steadily decreasing supply. This depletion is not just a numerical figure but represents the real-world scenario where oil, as a pivotal energy source, is being exhausted at an accelerating pace due to consistent consumption. The educational takeaway is clear: understanding the rate of depletion can guide discussions on sustainability and the urgency for alternative energy sources.

Recursive Mathematical Relationships

The exercise utilizes recursive mathematical relationships to model the annual increase in oil consumption and the depletion of oil reserves. Recursive relationships use previous values to determine the next one in a sequence. In this scenario, they are essential for projecting the oil consumption and remaining reserves into the future.

Each year’s consumption is based on the last year’s consumption increased by 5%: \( \text{Annual consumption}_{\text{next year}} = \text{Annual consumption}_{\text{current year}} \times (1 + 0.05) \). Similarly, the remaining reserves are the previous year’s reserves minus the consumption for that year: \( \text{Oil remaining}_{\text{next year}} = \text{Oil remaining}_{\text{current year}} - \text{Annual consumption}_{\text{current year}} \).

Students must understand this pattern to perform accurate calculations over multiple years, as the exercise requires. It reflects a powerful concept in mathematics that applies to various fields, including finance, population studies, and, as seen here, resource management.