Problem 25 - Applied Calculus

Question

If \(t\) is in years, and \(t=0\) is January \(1,2005,\) worldwide energy consumption, \(r,\) in quadrillion \(\left(10^{15}\right)\) BTUs per year, is modeled by $$ r=462 e^{0.019 t} $$ (a) Write a definite integral for the total energy use between the start of 2005 and the start of 2010 (b) Use the Fundamental Theorem of Calculus to evaluate the integral. Give units with your answer.

Step-by-Step Solution

Answer

The total energy use between 2005 and 2010 is approximately 2407.27 quadrillion BTUs.

Step 1: Identify the Bounds for the Integral

Since the problem is asking for the total energy use from the start of 2005 to the start of 2010, we need to determine the corresponding time values in the equation. Given that \( t = 0 \) corresponds to the start of 2005, the start of 2010 will be \( t = 5 \). Thus, the integral will be evaluated from 0 to 5.

Step 2: Set Up the Definite Integral

The definite integral we need to evaluate is:\[ \int_{0}^{5} 462 e^{0.019t} \, dt \]This represents the total energy consumption from the start of 2005 to the start of 2010 in quadrillion BTUs.

Step 3: Find the Antiderivative

The antiderivative of the function \(462 e^{0.019t} \) needs to be calculated. Using the rule \( \int e^{kx} \, dx = \frac{1}{k} e^{kx} + C \), the antiderivative becomes:\[ \frac{462}{0.019} e^{0.019t} \]

Step 4: Evaluate the Integral Using the Fundamental Theorem of Calculus

Using the antiderivative from the previous step, apply the Fundamental Theorem of Calculus:\[ \left[ \frac{462}{0.019} e^{0.019t} \right]_{0}^{5} \]Calculate the values:\[ \frac{462}{0.019} (e^{0.019 \times 5} - e^{0.019 \times 0}) \]

Step 5: Perform Calculations

Compute the numerical value:1. First determine the constant component: \( \frac{462}{0.019} \approx 24315.79 \).2. Compute exponential values: \( e^{0.095} \) using a calculator to approximate it as \( 1.099 \) (since \( 0.019 \times 5 = 0.095 \)).3. Compute the definite integral: \[ 24315.79(1.099 - 1) \approx 24315.79 \times 0.099 \approx 2407.27 \]This result is the total energy consumption from 2005 to 2010 in quadrillion BTUs.

Key Concepts

Exponential Functions

Exponential functions are vital in modeling growth processes, like energy consumption over time. These functions have the form \( f(t) = a e^{kt} \), where \( a \) is a constant, \( e \) is the base of the natural logarithm (approximately 2.718), and \( k \) is the growth rate. In our exercise, the function \( r = 462 e^{0.019t} \) models worldwide energy consumption, where 462 represents the initial amount of energy consumed, in quadrillion BTUs per year, and 0.019 is the annual growth rate.

As \( t \) increases, exponential functions grow at an increasing rate, making them suitable for calculating cumulative processes spread over time. This growth behavior is why they effectively represent real-world phenomena like compound interest or, as in our exercise, energy consumption.

In practical applications, understanding how to manipulate and integrate these functions is essential for predicting future trends and making informed decisions based on past data.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the gap between differentiation and integration, two core concepts in calculus. It tells us that if \( F \) is the antiderivative of a function \( f \), then the definite integral from \( a \) to \( b \) is given by \( F(b) - F(a) \). This theorem is essential when determining the total amount of a varying quantity, like our energy consumption over a specific time period.

In our exercise, the function \( r = 462 e^{0.019t} \) needs to be integrated over the interval from 0 to 5 (representing 2005 to 2010). By finding the antiderivative and applying this theorem, we can calculate the total energy used in that timeframe. Once we compute the antiderivative \( \frac{462}{0.019} e^{0.019t} \), we simply evaluate it at the bounds \( t = 5 \) and \( t = 0 \).

The powerful aspect of this theorem is it simplifies finding accumulated changes, revealing the total energy usage over several years with precise calculations.

Energy Consumption Model

Energy consumption models help us understand and predict how much energy will be used over time. These models can reveal patterns and are instrumental in planning for future energy needs or assessing the sustainability of current energy use.

The model given by \( r = 462 e^{0.019t} \) showcases exponential growth in energy consumption from 2005 onwards, which aligns with many countries' increasing energy demands. To quantify this consumption from 2005 to 2010, a definite integral is used, allowing us to express this cumulative use as a single, meaningful value in quadrillion BTUs.

Such models are crucial in various fields, including the design of policies for energy conservation and in the negotiation of energy contracts. They can also be used to evaluate the impact of different scenarios, like changes in efficiency or shifts in population growth.

• Real-world applicability ensures that these mathematical models are integral in both industries and government planning.
• Understanding these models helps stakeholders make informed decisions concerning energy production and usage.