Problem 13 - Calculus Concepts

Question

Energy Consumption (Historic) Between 1975 and 1980, energy consumption in the United States was increasing at an approximately constant rate of 1.08 quadrillion Btu per year. In 1980 , the United States consumed 76.0 quadrillion Btu. (Source: Based on data from Statistical Abstract, 1994\()\) a. Write a differential equation for the rate of change of energy consumption. b. Write a general solution for the differential equation. c. Determine the particular solution for energy consumption. d. Estimate the energy consumption in 1975 as well as the rate at which energy consumption was changing at that time.

Step-by-Step Solution

Answer

In 1975, energy consumption was 71.2 quadrillion Btu, with a constant rate of change of 1.08 quadrillion Btu/year.

Step 1: Define the Variables and Differential Equation

Let \( E(t) \) represent the energy consumption in quadrillion Btu at year \( t \). Since the rate of change of energy consumption is constant, the rate can be defined as \( \frac{dE}{dt} = 1.08 \).

Step 2: Write the General Solution to the Differential Equation

To find the general solution, integrate the differential equation \( \frac{dE}{dt} = 1.08 \). The integration yields \( E(t) = 1.08t + C \), where \( C \) is the integration constant.

Step 3: Determine the Particular Solution Using Initial Condition

In 1980, the energy consumption \( E(1980) = 76.0 \) quadrillion Btu. Substitute this into the equation: \[ 76.0 = 1.08 \times 1980 + C \].Solve for \( C \) to find \( C = 76.0 - 2138.4 = 119.6 \). Therefore, the particular solution is \( E(t) = 1.08t - 2059.6 \).

Step 4: Estimate Energy Consumption in 1975

To find the energy consumption in 1975, substitute \( t = 1975 \) into the particular solution:\[ E(1975) = 1.08 \times 1975 - 2059.6 \].Calculate to find \( E(1975) = 71.2 \) quadrillion Btu.

Step 5: Estimate the Rate at Which Energy Consumption Was Changing in 1975

Since the rate of change is constant, the rate at which energy consumption was changing in 1975 is simply the constant rate found earlier, which is 1.08 quadrillion Btu per year.

Key Concepts

Energy Consumption

Energy consumption refers to the total amount of energy used by an entity or group over a specific period. Here, we're looking at the United States from 1975 to 1980. During this period, energy consumption increased at a steady rate.
It is essential to grasp the historical context as energy policies, technologies, and economic factors influence consumption.
Understanding past trends helps predict future needs. In this case, knowing that the U.S had a constant increase in energy consumption by 1.08 quadrillion Btu per year provides insight into the growing energy demands during that period.
Such data can guide policymakers and industries in planning and managing resources effectively to ensure sustainable development.

Rate of Change

The rate of change is a measure of how much a quantity increases or decreases over time. In mathematical terms, it is often represented as a derivative.
For this exercise, the rate of change of energy consumption is given and constant at 1.08 quadrillion Btu per year.
- This implies that every additional year saw an increase in energy usage by this amount.- Understanding this, scientists and economists can anticipate the future energy demand if similar conditions persist.
In our context, the differential equation \( \frac{dE}{dt} = 1.08 \) mathematically describes this rate.
It is crucial as a foundational concept in understanding how small changes in time can have significant impacts on total energy consumption.

Particular Solution

A particular solution of a differential equation is a specific solution that satisfies not only the equation but also the initial or boundary conditions.
In simpler terms, it is like finding the puzzle piece that fits exactly into our specific scenario.
In the problem, we relied on the known energy consumption in the year 1980 to determine this unique solution.
By substituting the known value into our general equation \( E(t) = 1.08t + C \), we calculated the constants to find the particular solution: \( E(t) = 1.08t - 2059.6 \).
- This function now precisely models energy consumption over the period studied.- It allows us to estimate energy levels for any given year in the range of data we have, as demonstrated when calculating for 1975.

General Solution

In differential equations, the general solution is an expression that describes a family of functions that fulfills the differential equation for different initial conditions or constants.
The beauty of a general solution lies in its flexibility and universality. - It provides an overarching formula before initial conditions are applied, useful in various scenarios.
For the given rate of change, integrating the differential equation \( \frac{dE}{dt} = 1.08 \) reveals the general solution \( E(t) = 1.08t + C \).
- "C" here represents any constant that can vary based on additional information or conditions. - It's like having a skeleton key that opens multiple doors before you decide on one specific path.
Only by applying specific information, such as the data for the year 1980, do we narrow this down to a particular solution, fully customizing it to our historical scenario.
The general solution marks the starting point for solving real-world problems by providing the groundwork for more specific answers as new data becomes available.