Problem 50 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Because of the increasingly important role played by coal as a viable alternative energy source, the production of coal has been growing at the rate of $$ 3.5 e^{0.05 \mathrm{y}} $$ billion metric tons/year, \(t\) yr from 1980 (which corresponds to \(t=0\) ). Had it not been for the energy crisis, the rate of production of coal since 1980 might have been only $$ 3.5 e^{0.011} $$ billion metric tons/year, \(t\) yr from 1980 . Determine how much additional coal was produced between 1980 and the end of the century as an alternate energy source.

Step-by-Step Solution

Answer

The additional coal produced between 1980 and the end of the century as an alternate energy source is approximately 42.91 billion metric tons.

Step 1: Find the difference between the current and hypothetical rate of production

To find the difference between the current rate of coal production and the hypothetical rate of coal production without the energy crisis, subtract the hypothetical rate from the current rate. Difference = Current rate - Hypothetical rate = \(3.5e^{0.05t} - 3.5e^{0.01t}\)

Step 2: Set up the integral

Now, we need to integrate the difference of the two rates over the given time period (1980 to 2000). The integral will be: $$ \int_{0}^{20} (3.5e^{0.05t} - 3.5e^{0.01t}) dt $$

Step 3: Evaluate the integral

To evaluate the integral, split it into two separate integrals: $$ 3.5 \int_{0}^{20} e^{0.05t} dt - 3.5 \int_{0}^{20} e^{0.01t} dt $$ Now, integrate each part: $$ 3.5 [\frac{1}{0.05} e^{0.05t}]_{0}^{20} - 3.5 [\frac{1}{0.01} e^{0.01t}]_{0}^{20} $$

Step 4: Apply the limits

Next, apply the limits to each part of the integral: $$ 3.5 [\frac{1}{0.05} (e^{0.05(20)} - e^{0.05(0)})] - 3.5 [\frac{1}{0.01} (e^{0.01(20)} - e^{0.01(0)})] $$

Step 5: Simplify the expression

Finally, simplify the expression to find the total additional coal produced: $$ 3.5 [\frac{1}{0.05} (e^{1} - 1)] - 3.5 [\frac{1}{0.01} (e^{0.2} - 1)] $$ $$ \approx 3.5 (20e^1 - 20) - 3.5 (100e^{0.2} - 100) $$ $$ \approx 3.5 (20e - 20 -100e^{0.2} +100) $$ Now, calculate the value: $$ \approx 3.5(20(2.718) - 20 -100(1.221) +100) = 3.5(54.36 - 20 - 122.1 +100) \approx 3.5(12.26) \approx 42.91 $$ So, the additional coal produced between 1980 and the end of the century as an alternate energy source is approximately 42.91 billion metric tons.

Key Concepts

Integration

Integration is a key concept in calculus, often used to find the total accumulation of quantities. In the context of our problem about coal production, integration helps us calculate the total amount of coal produced over a specific time frame.

The process involves setting up an integral that represents the difference in production rates, which addresses how much more coal was produced due to the crises compared to a hypothetical scenario where it didn't occur. The fundamental idea is that through integration, we can measure these total differences over time by calculating the area under the curve of the production rate.

• The rate of production is modeled as an exponential function, which reflects how production grows over time.
• The exponential nature of the function hints at continuous growth, making integration the tool we use to find the accumulation over discrete intervals.
• The definite integral from 0 to 20 reflects the time from 1980 to 2000, two decades where this increased production took place.

By evaluating this integral, we determine how much additional coal was produced due to the shift in production trends in response to increased energy demand and alternative energy sources.

Exponential Growth

Exponential growth is a powerful concept that explains how quickly things can increase. It is characterized by a growth rate that becomes ever faster as time passes. In the case of coal production, exponential growth is reflected in the rate of production formula:

\[3.5 e^{0.05t}\]

• Here, the base (\(e\)) is the mathematical constant approximately equal to 2.718, symbolizing continuous growth.
• \(0.05t\) represents the exponent, where the growth rate is 5% per year compounding continuously from 1980.

This indicates not just a steady increase, but one that accelerates over time. If unchecked, exponential growth can lead to significant contrasts between what is and what could have been (as shown by the lower rate \(3.5 e^{0.01t}\)). By comparing these two scenarios, we can quantify how much the energy crisis increased coal production – not linearly, but exponentially over time.

Energy Production

Energy production is at the heart of economic and technological development. The significant role of coal in energy production during the 1980s was amplified due to increased demand. The use of coal as an alternative energy source cannot be overstated.

Several factors contribute to the surge in coal production:

• Coal served as a primary backup when other energy resources became less reliable or cost-effective due to the energy crisis.
• The readiness to increase coal production showcased its feasibility as a scalable solution to meet rising energy demands.

The additional coal produced acted as a buffer in the energy market, preventing potential shortages. This task of meeting energy demands underlined the importance of adaptive strategies in energy production that can leverage various sources according to availability and economic conditions. Thus, the calculated increase in coal production highlights not just a mathematical prediction but a real-world adjustment to global energy challenges.