Problem 9 - Applied Calculus

Question

World annual natural gas \(^{8}\) consumption, \(N,\) in millions of metric tons of oil equivalent, is approximated by \(N=\) \(1770+53 t,\) where \(t\) is in years since 1990 (a) How much natural gas was consumed in \(1990 ?\) In \(2010 ?\) (b) Estimate the total amount of natural gas consumed during the 20 -year period from 1990 to 2010 .

Step-by-Step Solution

Answer

1990: 1770 million tons; 2010: 2830 million tons; Total (1990-2010): 46,000 million tons.

Step 1: Define the formula for natural gas consumption

The exercise provides the formula for natural gas consumption as \( N = 1770 + 53t \), where \( N \) is the consumption in millions of metric tons of oil equivalent and \( t \) is the number of years since 1990.

Step 2: Calculate consumption for 1990

In 1990, \( t = 0 \) since it is the base year. Substitute \( t = 0 \) into the formula to find \( N \):\[ N = 1770 + 53(0) = 1770 \]Thus, the consumption in 1990 was 1770 million metric tons of oil equivalent.

Step 3: Calculate consumption for 2010

For 2010, \( t = 20 \) because 2010 is 20 years after 1990. Substitute \( t = 20 \) into the formula:\[ N = 1770 + 53(20) = 1770 + 1060 = 2830 \]Thus, the consumption in 2010 was 2830 million metric tons of oil equivalent.

Step 4: Calculate the total consumption over 20 years

To find the total consumption from 1990 to 2010, determine the total increase over the period and then calculate the series sum. Since there is a linear increase:Total increase over 20 years is \( 53 \times 20 = 1060 \).The average consumption over the period is the midpoint between the values at 1990 and 2010:\[ \text{Average Consumption} = \frac{1770 + 2830}{2} = 2300 \]Multiply by the number of years:\[ \text{Total Consumption} = 2300 \times 20 = 46000 \]The total consumption over the 20-year period was 46,000 million metric tons of oil equivalent.

Key Concepts

Linear Growth Formula

In the context of estimating natural gas consumption, the linear growth formula is a powerful tool. It provides a simple way to project future values based on a consistent rate of increase or decrease. The given formula,\[ N = 1770 + 53t \]illustrates this concept well. Here, \( N \) represents the natural gas consumption in millions of metric tons of oil equivalent, and \( t \) corresponds to the number of years since the base year, 1990. The components of the formula have specific meanings:

• **1770** is the initial amount of natural gas consumption in the baseline year, 1990.
• **53** acts as the constant rate of increase, indicating that every year after 1990, consumption grows by 53 million metric tons.

This linear approach is useful because it allows predictions and insights simply by substituting different values of \( t \) to reflect the passing years. It helps us understand both current and future consumption patterns in an intuitive manner.

Calculation of Annual Consumption

When applying the linear growth formula to estimate natural gas consumption for specific years, the calculation is straightforward and insightful. Let's break it down:

Year 1990 Calculation

Since 1990 is the base year, we set \( t = 0 \). Substituting into the formula:\[N = 1770 + 53 \times 0 = 1770\]This means, in 1990, the consumption was 1770 million metric tons of oil equivalent.

Year 2010 Calculation

For the year 2010, which is 20 years after the base year, \( t = 20 \). Substituting this into the formula gives:\[N = 1770 + 53 \times 20 = 2830\]Thus, in 2010, the consumption rose to 2830 million metric tons. These calculations are pivotal in demonstrating how the formula enables us to determine annual consumption effortlessly.

Total Consumption Estimation

Estimating total consumption over a period, such as from 1990 to 2010, involves integrating the linear growth into a cumulative perspective. The first step is to identify the total increase over this time:

• The total increase is calculated as \( 53 \times 20 = 1060 \), representing the consistent yearly increments over 20 years.

Next, we find the average consumption across the period. The average is given by:\[\text{Average Consumption} = \frac{1770 + 2830}{2} = 2300\]This average value captures the typical consumption rate throughout the years. Finally, to estimate the total consumption, we multiply this average by the number of years:\[\text{Total Consumption} = 2300 \times 20 = 46000\]This computation reveals that over the 20-year span, the total natural gas consumed amounted to 46,000 million metric tons. This holistic view of consumption helps in understanding the broader trends and patterns, facilitating informed planning and decision-making.